Semigroups of Linear and Nonlinear Operations and Applications: Proceedings of the Cura§ao...

SEMIGROUPS OF LINEAR AND NONLINEAR OPERATIONS AND APPLICATIONS

Semigroups of Linear and N onlinear Operations and Applications Proceedings of the Cura~ao Conference, August 1992

Edited by

GISELE RUIZ GOLDSTEIN

and

JEROME A. GOLDSTEIN Department of Mathematics, Louisiana State University Baton Rouge, Louisiana, U.SA

SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

A c.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-4834-7 ISBN 978-94-011-1888-0 (eBook) DOI 10.1007/978-94-011-1888-0

Printed an acid-free paper

All Rights Reserved © 1993 Springer Science+Business Media Dordrecht Origina1ly published by Kluwer Academic Publishers in 1993 Softcover reprint of the hardcover 1 st edition 1993 No part of the material protected by this copyright notice may be reproduced ar utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information starage and retrieval system, without written permission from the copyright owner.

Table of Contents

Director's Preface

Editors' Preface

Jerome A. Goldstein: A Survey of Semigroups of Linear Operators and Applications

Gisele Ruiz Goldstein:

3

7

9

Nonlinear Semigroups and Applications 59

Wim Caspers and Philippe Clement: A Bifucation Problem for Point Interactions in Ll (IR?) 99

Alfonso Castro and Ratnasingham Shivaji: Semipositone Problems 109

Iona Cioranescu: A Generation Result for C-regularized Semigroups 121

Giuseppe Da Prato: Smoothing Properties of Heat Semi groups in Infinite Dimensions 130

W.E. Fitzgibbon, S.L. Hollis, and J.J. Morgan: Locally Stable Dynamics for Reaction-Diffusion Systems 143

W. Fitzgibbon, M. Parrott, and Y. You: Global Dynamics of Singularly Perturbed Hodgkin-Huxley Equations 159

Matthias Hieber: On Strongly Elliptic Differential Operators on Ll(JRn ) 177

Alessandra Lunardi: Stability and Local Invariant Manifolds in Fully Nonlinear Parabolic Equations

Adam C. McBride: Fractional Integrals and Semigroups

Rainer Nagel: Spectral and Asymptotic Properties of Strongly Continuous Semigroups

1

185

205

225

2 Table of Contents

J. W. Neuberger: Continuation for Quasiholomorphic Semigroups

Michel Pierre and Didier Schmitt: Global Existence for a Reaction-Diffusion System with a Balance Law

G. F. Webb: Convexity of the Growth Bound of Co-semigroups of Operators

Index

241

251

259

271

Director's Preface

This is the first publication which follows an agreement by Kluwer Publishers with the Caribbean Mathematics Foundation (CMF), to publish the proceedings of its mathematical activities. To which one should add a disclaimer of sorts, namely that this volume is not the first in a series, because it is not first, and because neither party to the agreement construes these publications as elements of a series. Like the work of CMF, the arrangement between it and Kluwer Publishers, evolved gradually, empirically.

CMF was created in 1988, and inaugurated with a conference on Ordered Algebraic Structures. Every year since there have been gatherings on a variety of mathematical topics: Locales and Topological Groups in 1989; Positive Operators in 1990; Finite Geometry and Abelian Groups in 1991; Semigroups of Operators last year. It should be stressed, however that in preparing for the first conference, there was no plan which might have augured what came after. One could say that one thing led to another, and one would be right enough.

Yet, that is not quite the whole story, For a long time it had been clear to me that the Caribbean Basin and Latin America languished in a general state of mathematical oblivion. And having said that I'm reminded that one should be careful with such statements, that there certainly are a number of distinguished mathematicians from this part of the world, and also a number of centers where first-rate work is being carried out. That is not the point, however; what is the raison d'etre of CMF is that the men and women and centers of distinction in mathematics in the Caribbean Region and Latin America are few and far between; that the vast majority of mathematicians there gradually lose contact with the developed world because they are isolated in underdevelopment, cut of from physical contact with experts. They are isolated because their libraries are persistently out of date.

CMF enters this stage of underdevelopment with a mission which is easy to formulate and yet difficult to carry out. The mission is this: to do all it can to diminish the isolation alluded

3

G. R. Goldstein and J. A. Goldstein (eds.), Semigroups of Linear and Nonlinear Operations and Applications, 3-6. © 1993 Kluwer Academic Publishers.

4 Director's Preface

to in the preceding paragraph. It is a difficult task for two main reasons. First, it is not always clear how one can do the most good. (More on this subject shortly.) And even when it is, there are problems, faced by mathematicians of the region, which are so typical of underdevelopment: poverty, institutional paternalism and bureaucracy. We, in our cozy, developed universe, who complain about the shortage of funds when it comes to travel to conferences, should have some idea of the shortage faced by mathematicians in the so-called Third World. As it seems inappropriate to use these pages to compile woes, let me illustrate with one example: a mathematician from EI Salvador, Nicaragua, Surinam or Guyana will typically get no funds at all from his/her institution or home country, even when there is an explicit letter of invitation.

But perhaps one need not strain to document poverty in underdevelopment. Institutional paternalism, however, is less palpable, and the effects of bureaucracy certainly as cruel as those of poverty. The Spanish have a word for the kind of "red tape chase" that takes days, weeks, months, and saps one's enthusiam: tramite. One Colombian colleague recently quipped that his country was the land of "tramitologia".

In the developed world a mathematician who has made plans to attend a conference simply arranges for a colleague to take over his classes for a few days and leaves. Not so in the Caribbean Region or Latin America, in general. There one has to submit a formal request for leave, which then travels through the entire bureaucratic apparatus of a university. If an academic goes to a conference without official permission, he runs a great risk of not being paid for the time away.

Such are the problems, even in the presence of the best laid plans. The good news is that there are enough mathematicians in the developed world with a true missionary spirit to challenge the condition of underdevelopment in this region. It helps to be able to invite one's colleagues to a tropical island like Cura~ao, where the swimming and snorkeling are quite satisfactory, and where the Dutch fa~ades on the St. Anna Bay strike the typical visitor with more than a touch of wonder. And yet, not every distinguished mathematician is suited to confront the special problems of underdevelopment. Even the most altruistic soul has seen the reality and been disappointed: at the low attendance by Caribbean and

Director's Preface 5

Latin American mathematicians at the workshops. I have seen the disappointment in the faces of my colleagues, and shared in it.

Which brings me round to the first-mentioned problem, in discharging the simple mission of CMF. Deciding what themes and topics to choose for the workshops and conferences.

Doubtless the attendance at these events by mathematicians from the region is small because of the poverty and bureaucracy to which I have already referred. However, one should also realize that the region in question is huge, and that the interests of its mathematicians varies enormously, not only in terms of discipline or speciality, but also in terms of level of development. Last year's workshop/conference on Semigroups of Operators and Applications is a good case in point. The conference attracted four South American mathematicians: two from Chile and two from Colombia. One might stare at that figure and say it is small. My experience tells me that to draw four mathematicians from South America constitutes an enormous success. The workshop portion of the program drew another three participants from Surinam and one Jamaican. What this number does not reveal is that the latter four have become regulars at these events. Two of the Surinamers have been attending since 1990, the third since 1991, the Jamaican since the beginning.

One comes to an attitude of compromise and resignation, while not losing sight of the goal, as stated above. Missionaries of true spirit know how frustrating the task of promulgating the Gospel can be. One has to know that the job is worth doing: it is good to disseminate mathematics in the Caribbean Region and Latin America. In the meantime one tries to find out how best to serve its mathematical community, knowing that there will be moments of frustration along the way; that one will make mistakes along the way. One has to know that the crowds will be small; that the participants who return, year after year, for whatever reason, are, in the long run, the best promotors one has.

One counts one's blessings; the greatest two being to be able to rely on the services of colleagues who instinctively understand that the problem of doing something about underdevelopment is not so simple, yet who by their talent and spirit contribute to the realization of CMF's goals, with excellent programs. No one has done this with more grace and verve than Jerry Goldstein and

6 Director's Preface

Gisele Ruiz Goldstein, through their program on Semigroups of Operators and Applications. Generally, I have a solid expectation that the colleagues with whom I contract to prepare a workshop/conference for CMF will acquit themselves well. Indeed, I can be more bold: I count on the people selected to present a program under the auspices of CMF, which will produce an event of the highest quality, and also display the kind of sensitivity one needs when coming face to face with underdevelopment. I have not yet been disappointed on either point.

During the three days of the workshop, the Goldsteins presented an account of both the linear and non-linear theories of semi groups of operators, by clearly grounding the subject both in terms of its origins in physics and of the philosophy of its development, returning with regularity to a number of illustrative examples. Perhaps it is not quite fair to volunteer myself as a judge, but, as decidedly a non-expert in this field, I was able to see the logic from formulation to execution, and understand, if very few of the details, at least the mathematical tools that are brought to bear. It seems to me that one cannot ask more of such a workshop presentation.

As to the conference which followed, I can only judge by the comments of its participants. I spoke to almost all of the twenty or so, and each was thoroughly satisfied with the mathematical event. We plan to reprise a conference on this discipline before the end of the decade.

I spoke earlier of two blessings; I've expatiated on one, but have not forgotten the other. One is fortunate to have in Dr. David Larner and his staff at the Science and Technology Division of Kluwer Academic Publishers people who have understood the problem of disseminating mathematics in the region, and have backed this understanding and their encouragement of CMF with deeds. Their agreement to be the publishers of these proceedings is but one of them.

Jorge Martinez, Director

Caribbean Mathematics Foundation

Gainesville, FL.

April 1993

Editors' Preface

The conference on Semi groups of Operators and Applications took place in Curac;ao during August 1992. In the first week the two editors gave short courses to the Caribbean Mathematics Foundation, Gisele Goldstein's lectures being on nonlinear semigroups and Jerry Goldstein's being on linear semigroups. Those short courses sketched the theory and emphasized the applications, especially to PDE. The second week was devoted to an international conference, and lectures were given by some of the leading people in the field.

Curac;ao is a paradise, and everyone, especially the undersigned, was astonished by the breathtaking beauty of the island. Jorge Martinez, the Director of the Caribbean Mathematical Foundation, is a masterful organizer. From his choice of hotel and conference location through the elegant coffee breaks to the conference banquet and other entertainment, he was the perfect host. In particular, he arranged for perfect weather throughout the meeting. We were all touched by Jorge's love for mathematics and for the Caribbean region. All the participants had a wonderful time.

The first two papers are written versions of the short courses given by the editors. The audience was lively and attentive and we are grateful for that. The rest of the papers are from the international conference portion of the program.

Some of the lectures given at the conference appear here as jointly authored papers. The corresponding lectures were presented by Alfonso Castro, Philippe Clement, Bill Fitzgibbon, Mary Parrott, and Michel Pierre.

Also, several interesting lectures were given which do not appear in these written proceedings. Those were:

J. Robert Dorroh: "Existence and regularity of solutions of singular quasi-linear diffusion equations",

Klaus Engel: "On dissipative wave equations in reflexive Banach spaces" ,

7

G. R. Goldstein and 1. A. Goldstein (eds.), Semigroups of Linear and Nonlinear Operations and Applications, 7-8. © 1993 Kluwer Academic Publishers.

8 Editors' Preface

Hernan Henriquez: "Periodic solutions of quasi-linear partial functional differential equations with unbounded delay" ,

Pedro Isaza J. and Jorge E. Mejia L. : "Local solutions of the Kadomstev-Petviashvili equation with periodic conditions",

Enzo Mitidieri: "Blow-up estimates for semilinear parabolic systems",

Frank Neubrander: "Degenerate abstract Cauchy problems",

Andreas Stahel: "Equations of a vibrating plate" ,

Silvia Teleman: "One subspace" ,

Ricardo Weder: "Obstacle scattering without local compactness".

The Caribbean and Latin Americans present at the conference were Halvard White of Jamaica, H. Antonius, C. Gorisson and Henrietta Ilahi of Surinam, P. Isaza and J. Mejia of Columbia, H. Henriquez and C. Lizama of Chile, 1. Cioranescu and S. Teleman of Puerto Rico, and R. Weder of Mexico.

We are grateful to two members of the LSU staff who cheerfully and efficiently did much typing for us. Susan Oncal handled the correspondence and J acquie Paxton did the technical typing.

Finally we are grateful to NSF for their support of our research.

Gisele Ruiz Goldstein

Jerome A. Goldstein

Baton Rouge, LA.

April 1993

A Survey of Semigroups of Linear Operators and Applications

Jerome A. Goldstein

One parameter semigroups of linear operators have a simple ele

gant theory together with a wide variety of applications, covering

a broad area. These lecture notes give an introduction to this

theory and indicate some of the applications.

The emphasis here is on the extraordinary variety of appli

cations. The presentation will be breezy. Proofs will usually be

replaced by sketches or heuristic arguments.

Our starting point is the abstract Cauchy problem (or ab

stract initial value problem)

du =Au dt

(t 2:: 0), u(o) = f. (ACP)

Here u maps R+ = [0,(0) into a space X, and the term "abstract"

indicates that X is taken to be a Banach space, usually infinite

dimensional. The meaning of du/ dt = Au is that

u(t + h) - u(t) _ A(u(t)) ---+ 0 h

as h ---+ o. Thus we want to take linear combinations and limits

of members of X; taking X to be a Banach space is a natural as

sumption. The operator A: V(A) eX ---+ X is linear. The initial

condition u(O) = f makes sense if one views u as a continuous

curve in X.

9

G. R. Goldstein and J. A. Goldstein ( eds.), Semigroups of Linear and Nonlinear Operations and Applications, 9-57. © 1993 Kluwer Academic Publishers.

10 1. A. Goldstein

The problem (ACP) is well posed if a solution exists, is

unique, and depends continuously on the ingredients of the prob

lem ( namely I and A). The well-posed problems are the ones

which interest us in this course.

Uniqueness allows us to express the solution at time t + s in

two ways. One is as u( t + s) = T( t + s )/, where T( T) maps the

solution at some time to the solution T units of time later. (T( T)

only depends on the length T of the time interval and not on its

end points since the equation is autonomous, i.e., since A does

not depend on the time t.) One can also write u(t + s) = T(t)g,

where 9 = T( s ) I is the solution at time s. Thus

T(t + s)1 = T(t)(T(s)/).

Since each T( T) should be continuous and since we may assume

that V( A) is dense in X (otherwise replace X by V( A) ), we are

led to the following definition. T = {T(t) : t E 1R+} is a strongly

continuous one parameter semigroup of operators on X (or simply

a (Co) semigroup on X) if

T(t + s) = T(t)T(s) for all t,s ~ 0,

T(O) = I,

T(')I E C(1R+, X) for all I E X,

i.e., t -+ T(t)1 is continuous from 1R+ to X for all lEX. Starting

with a (Co) semigroup T, one can associate an operator A and

a corresponding (ACP) with T as follows. The (infinitesimal)

generator of T is A = T'(O). More precisely, I is in the domain of

A, V(A), iff lim T(h>f-f exists; then AI is the limit. h-+O

Formally T(t) = etA, and

Semigroups of Linear Operators 11

Suppose T is a (Co) contraction semigroup, i.e. a (Co) semigroup

satisfying IIT(t)11 ::; 1 for all t > O. It follows from the above

formal calculation that

for all A > 0, II(AI - A)-III ::; ~ (2)

and

JR(AI - A) = X for all A > O. (3)

Here JR(B) is the range of an operator B. Setting a = l (2) is ,

equivalent to

for all a > 0, II(I - aA)-lll ::; 1. (2')

An operator satisfying (2) [ or (2')] is called dissipative; an oper

ator satisfying (2) and (3) is called m-dissipative.

HILLE- YOSIDA THEOREM.

A n operator A is the generator of a ( Co) contraction semi

group T = {T(t) : t E JR+} iffV(A) = X and A is m-dissipative.

In this case (ACP) is well-posed; if f E V(A), the unique solution

u E C (JR+, V(A)) n C1(JR+, X) is given by u( t) = T(t)f. Finally,

(repres entation)

T(t)f = lim (I - !A)-n f for all t, fi n--+oo n

(regularity)

T(t)('D(An)) C V (An) for all t E JR+ and n E IN = {I, 2, 3, ... },

and if f E V(An), T(·)f E C (JR+, V(An)) n cn(JR+,x).

12 J. A. Goldstein

The proof is not difficult. The necessity follows easily from

the variant of (1) with a vector in it to make the integrals exist:

100 e->.tT(t)fdt = ()"I - A)-l f

for f E X and)" > O. Thus ()"I - A)-l is the Laplace transform

of the semigroup T. One way to prove the sufficiency is to invert

the Laplace transform. Many other proofs are known.

The main examples come from partial differential equations

(PDE):

heat equation

wave equation

au/at = .6.u

a2u/at2 = .6.u

Schrodinger equation iau/at = .6.u.

Consider the heat equation for t ~ 0 with x E n, a smooth

domain in IRn. Associate with the Laplacian .6. the homogeneous

Dirichlet boundary condition u = 0 for x E an ( = boundary

of n) and all t ~ O. Let X be a space of functions on n (e.g.

LP(n),l ::; p < 00). Let A be the (distributional) Laplacian

acting on DCA) = {f E X n C(s1) : .6.f E X, flar.! = O}. By

identifying x -t u(t,x) with u(t) E X, the problems

and

au/at = .6.u in IR+ x n,

u(x,O) = f(x) in n,

u ( x, t) = 0 on IR + x an

du(t)/dt = Au(t)

u(O) = f

are formally equivalent. In this case, either A or the closure A of

A is densely defined and m-dissipative. A similar result holds for

the Schrodinger equation in L2. To make the wave equation look

like (ACP), simply reduce it to a first order system: Let

v = ( ~~) = (au/at) , A = (~ ~). Then Utt = ~u is formally equivalent to dv/dt = Avon a space

of pairs of functions on n. Notice that boundary conditions can

be present in (ACP); they are built into the description of 'D(A).

SIMILARITY.

Let X, Y be Banach spaces. Let U : X -+ Y be linear, bijec

tive and bicontinuous. Let A = U AU-I, T(t) = UT(t)U- 1 .

Then A generates the (Co) semigroup T on X iff A gener

ates the ( Co) semi group T on Y. An example of such a U

is the Fourier transform F, which is unitary from L2 (lR~) to

L2 (lRe). Thus, for 1 = Ff, 1(0 = (27r)-n j 2 fJR n e-ix.~ f(x)dx

(for f E L2(lR~)nLl(lR~)). For aa = a~la;2 ... a~n,aj = a / ax j, (aa ft (0 = (ioa 1( 0; thus F converts differential oper-

n ators into multiplication operators. lal = I: a j is the order of

j=l

Let X = L 2 (n,"E,,/-l) and let m : n -+ <r be "E,

measurable. The operator of multiplication by m is Mm defined

by (Mmf)(w) = m(w)f(w), w E n,andf E 'D(Mm)ifff,mf E X.

Let m(e) = - clel 2 , X = L2 (lRn). For X = L2(lR) let v(O = ie.

Then

(etdjdx f)(x) = f(x + t),

d/dx = F-1 MvF,

etdjdx = F-1M F e tv ,

14 1. A. Goldstein

When c = 1, this gives the formula for the solution of (ACP) for

the heat equation on IRn with t > 0 (or Re t > 0). When c = -i,

this gives the solution of (ACP) for the Schrodinger equation for

t E IR. In the latter case T is a (Co) unitary group, i.e. each

T(t) = e-it~ is a unitary operator on L2(JRn ). In the former case

{T(t) : Re t ~ O} is analytic in the right half plane {Re t > OJ.

ANALYTIC SEMIGROUPS.

Let L: be the pictured shaded sector in the complex plane.

Let A satisfy L: C p(A), the resolvent set. of A, and 11(,\1-

A)-III ~ ,M/I,\I for some AI> 0 and all ,\ E L:.

Semigroups of Linear Operators

Then

T(t)J = ~ { eAt(>..J - A)-l Jdt 27rZ ir

15

defines a (Go) semi group ( if V(A) = X) analytic in some sector

including the positive real numbers. Moreover

T(t)(X) C V(Am) for all m E IN.

In the case of the heat equation on IRn (see (4) with c = 1),

for all J E L2(IRn), t > 0, 00 00

u(t, .) = T(t)J E n V(Am) = n H2m(IRn) C Goo(IRn), m=l m=l

and

HILBERT SPACE RESULTS.

When X = H is a Hilbert space, A is dissipative iff

Re{AJ, f) ~ 0 for all J E V(A). [A similar result using semi

inner products or duality maps holds in a general Banach space.]

Let B : V(B) c H -t H be densely defined. Then g E V(B*)

and B* g = h means: there is a constant G such that

I{BJ,g)1 ~ GIIJII

holds for all J E V(B). Then the bounded linear functional J --+

(BJ,g) is given by (j,h)(= (j,B*g)). The denseness of V(B)

implies that h is uniquely determined.

For V(A) dense in H, A is symmetric if (AJg) = (j, Ag) for

all J,g E V(A) iff A c A*. A is selfadjoint if A = A*, i.e. A is

symmetric and V(A*) = V(A). The next two theorems can be

16 1. A. Goldstein

proved as a consequence of the Hille-Yosida theorem. We take

the Hilbert space to be complex.

STONE'S THEOREM.

A generates a (Co) unitary group on 1{ iffiA is selfadjoint.

SPECTRAL THEOREM.

(Hilbert, Stone, von Neumann). A on 1{ is selfadjoint iff

there is a measure space (D, L;, Il), a unitary map U : 1{ --+

L2(D,L;,Il), and a L;-measurable real function m: D --+ IR such

that

In this case, the spectrum a(A) of A is the closure of the

(essential) range of m. For all measurable functions F from a(A)

to <c, F(A) = U- 1 MnU is well defined, where n = F(m). In

particular ei tA = U- 1 Me i tm U.

The wave equation on IRn ,

Utt=6u (XEIRn , tEIR),

can be written as

Applying the Fourier transform F (in the space variables) gives

A conserved quality is the energy E given by

E = Ilutll~2 + 111~lu 11~2 = (Ut, Ut) + (1~12u, u) = Ilutll~2 + II( -6)1/2ull~2

= Ilut(t)lli2 + IIVxu(t)lli2'

Semi groups of Linear Operators 17

The fact that dE / dt == 0 leads to the conclusion that the wave

equation is governed by a (Co) unitary group on

1i = {(:~) : IIw211i2 + 1I(-~)1/2wllli2 < oo};

and E 1/ 2 is called the energy norm of the solution u.

Roughly speaking, parabolic equations (such as the heat equa

tion) are governed by analytic semigroups. Hyperbolic equations

(such as the wave equation) are governed by groups, i.e., they

are well-posed in forward time and in backward time. From this

point of view, the Schrodinger equation is hyperbolic. Some equa

tions (such as many functional differential equations) are neither

parabolic nor hyperbolic; they are governed by (Co) semigroups

which are neither analytic nor groups.

PERTURBATION THEOREMS.

1. Let A generate a (Co) contraction semigroup on X, i.e., A

is densely defined and m-dissipative. Let B be dissipative with

V(B) :J V(A) and suppose these exist constants a < 1, b 2: 0

such that

liB III ~ allAIIi + bllill (5)

holds for each I E V(A). Then A + B (defined on V(A)) is

m-dissipative.

This is a variant of a perturbation theorem about selfadjoint

operators, due to Kato and Rellich.

II. Let A be selfadjoint on 1i. Let B be symmetric with V(B) :J

V(A) and suppose that (5) holds, with a < 1. Then A + B ( on

V(A)) is selfadjoint; and A + B is semibounded if A is.

18 J. A. Goldstein

III. Let A generate an analytic semigroup on X and let D(B) :J

D(A). Suppose that there is a sufficiently small a > 0 and some

b > 0 such that (5) holds. Then A + B ( on D( A) ) generates an

analytic semigroup on X.

The "sufficiently small" part of the statement of Theorem III

is vague. In the applications, one can usually show that given any

C > 0, there is a Ce > 0 such that one can take a = c, b = Ce in

Theorems I, II, or III.

Here is an example. Let A = ~ or more generally let

A L: aaf38a+f3. Here each aaf3 is real and aaf3 = af3a lal=If3I=m

is assumed. On 11 = L2(JRn ),

(Au,u) = L (-1)m aaf31 8 au8f3u dx.

lal=If3I=m IRn

To insure that this is nonpositive we assume the ellipticity hy

pothesis: There is an Co > 0 such that

L (-l)maaf3~ae:::; -col~12m lal=If3I=m

for each ~ E JR n. Here ~a = ~~1 ~;2 ... ~~n, as usual. Thus ~

is elliptic with m = 1, ajk = 8jk , and co = 1. By the spectral

theorem and the associated operational calculus, A is a nonposi

tive selfadjoint operator and A generates a semi group analytic in

the right half plane. The last sentence holds for A + B as well

provided

B= 0:5laI9m- 1

where each ba E £<Xl(JRn ). The proof is as follows. Given k E

{O, 1, ... , 2m - I}, for each c > 0 there is a C = C ( c; 2m, n) > 0

such that

L Iloa fll~2 ~ fllAfll~2 + Cllfll~2' lal=k

This is easy to see using Fourier transforms together with

max 1~lk ~ fl~12m + C(f; 2m, n), O<k<2m-l

which holds for all ~ E lRn. Since each ba is bounded, Perturbation

theorem III applies to finish the proof.

The same results holds if lR n is replaced by n c lR n. In this

case one must impose boundary conditions to make A selfadjoint

and nonnegative. More generally A can have variable coefficients

(i.e., the aa(3 can depend on x E n) and one can work in the space

LP(n), 1 < p < 00. Solving u -'\Au = h (or u - '\(A+B)u = h)

and estimating in terms of ,\ and h is done using the theory of

elliptic boundary value problems.

We summarize. It is easy to construct examples of parabolic

PDEs governed by analytic semigroups on X = L 2 (n), with n = lRn, A a constant coefficient elliptic operator and B an operator

with lower order terms and bounded coefficients. Analogous ex

tentions to n c lR n, X = LP (n) and variable coefficients for A

all hold but are technically much more complicated and rely on

deep facts from linear elliptic PDE theory.

Similar results hold for wave equations of the form

Utt + (A + B)u = 0

using Perturbation theorem II, with A

before and with

L: aa(30a+(3 as lal=I(3I=m

20

Bu = L (_1)101+1,8180 [bo,8(x)8,8u] 101,1,8I:5 m - 1

1. A. Goldstein

with bo,8 complex valued, ba ,8 = b,8o, and 8"Ybo,8 E L2(IRn) for

each, with 1,1::; max{lal, 1,81}. This includes

n

Utt = ~u + L bj(x)8uj8xj + c(x)u j=1

for x E IR n, t E IR for certain choices of bl , ... , bn , c.

If C = A + B is positive and selfadjoint, then the energy

norm associated with the unitary group governing Utt + Cu = 0

is E I / 2 where ,

APPROXIMATION THEOREM.

Let An generate a (Co) contraction semigroup Tn on X for

n = 0,1,2, .... Then Tn(t)f -t To(t)f holds for all t 2: 0 and all

I in X iff (.\ - An)-I I -t (.\ - AO)-I I holds for all .\ > 0 and

all lEX. Sufficient (but not necessary) is that D(Ao) C 1)(An)

and Ani -t Aol holds for all f E 1)(Ao).

A simple special case is as follows: Let An E 8(X) (the

bounded linear operators on X) and An be dissipative for all

n 2:: 1, Ani -t Aol for all I E 1), where 1) is a core for A o, i.e.

Ao is the closure of its restriction to 1). Then the above conclusion

holds. We give a simple proof of this in the event that An and

To(t) commute for all t 2:: 0 and all n 2:: 1. For I E 1),


t d IITn(t)f - To(t)fll = 11- 10 ds Tn(t - s)To(s)fdsll

= lilt Tn(t - s)(An - Ao)To(s)fdsll

::; lilt IITn(t - s)To(s) (An - Ao) fllds

by the commutativity assumption

::; It''(An-Ao)f''ds

= tllAnf - Aofll-t 0

21

as n -t 00, uniformly for t in bounded sets in IR +. This last

conclusion holds in general in the context of the approximation

theorem.

EXAMPLE.

Let X = BUC(IR), the bounded uniformly continuous func

tions on IR, which is a Banach space under the supremum norm.

Ao = d/dx with V(Ao) = {f EX: f E C1(IR), l' EX}

generates the (Co) contraction semigroup To given by

(To ( t) f) ( x) = f (x + t).

(Cf. (4).) Let An = [(T(~) - I)/(l/n)] , for n E IN. Then,

uniformly for x E IR and t E [0, 1],

f(x + t) = lim lim ~ tm (A;:' 1) (x). n ..... = M ..... = L.....J m!

Letting x = 0 yields

m=O

Mm

f(t) = lim "tmCmn n~oo L.....J m=O

22 1. A. Goldstein

with Cmn = (A~ f) (O)jm! and Mm suitably chosen; this holds

uniformly. Thus any I E C[O, 1] ( which can be trivally extended

to be in X) is a uniform limit of polynomials; this is nothing but

the Weierstrass approximation theorem. A simple modification

of this argument yields a deeper and more interesting result due

to Hille. Consider the difference operator !::l.h defined by

(!::l.hf) (x) = I(x + h) - I(x). h

Then the approximation theorem (i.e. the version with h --+ 0

replacing n --+ (0) yields

M t m

I(x + t) = lim lim '" -, (!::l.h f) (x), (6) h->O M->oo ~ m.

m=l

uniformly for x E IR and t is bounded intervals, for all lEX.

Now look at (6) with the limits reversed:

M t m

I( x + t) = lim lim L -, (!::l. h f) (x). M->oo h->O m.

m=O

This is of course Taylor's theorem and it has many hypothe

ses, the main one being that I is analytic on IR. But the form

(6) works without any differentiability hypothesis! If one wants M m E !n! (!::l.h f) (x) to be a good approximation to I(x +t) for gen-

m=O eral continuous I, one should fix h i= 0 and take many terms; that

is M should get large before h gets small.

CHERNOFF FORMULA.

Let V(·) : IR+ --+ SeX) satisfy V(O) = I, 1!V(t)11 ~ 1 for

each t ~ 0, VOl E C(IR+, X) for each I E X, and V'(O) = A

on V where Alv = A generates a (Co) semigroup T. (The last


condition can be restated as V(h~f- f -+ AI for all 1 in a core 'D

of A.) Then

lim V (~) n 1 = T(t)1 n---+oo n

for all t ~ 0 and 1 E x.

The convergence is uniform for t in bounded sets, but that

will not concern us here.

Lemma. Let L E 8(X) with IILII ::::; 1. Then for all E x,

Ile-n(L-I)f - Lnlll::::; ynllLI - 111.

We omit the elementary proof.

N ow let the hypotheses of the Chernoff formula hold and take

( / V(i)-I

L = V t n), An = L . Then for a > 0, n

since IILII ::::; 1. Thus An is m-dissipative. For 1 E 'D,

v(~)n f-T(t)f={Lnf-en(L-I)f}

+ {en(L-I) 1 - T(t)f} == h + h.

Ilhll::::; ynllLI - 111 = ~ynll V(~y -111-+ 0 n t n

as n -+ 00 by the lemma, since the above expression approxi

mately equals

Next,

Ilhll = lIe tAn 1 - T(t)fll -+ 0

24 J. A. Goldstein

as n ---+ 00 by the approximation theorem, since Anf ---+ Af on 1).

The Chernoff formula follows. o

CENTRAL LIMIT THEOREM.

A probability space is a measure space (D,~, P) satisfying

P(D) = 1. A random variable is a ~-measurable function e : D ---+

JR. The distribution function F of ~ is given by

Fe(t) = P[~ ~ t] = p{e- 1 ( -00, t]}, t E JR.

Let 1):.F be the set of all distribution functions, i.e., the set of

all monotone nondecreasing F : JR ---+ JR which are continuous

from the right and satisfy lim F( t) = 0, lim F( t) = 1. Let t ---+ - 00 t ---+ 00

X = BUG(JR), the bounded uniformly continuous functions on

JR. Define the "tilde transform" to be the map

from 1):.F to 8(X) given by

(Ff)(t) = 1: f(t - s)dF(s), f E x, t E JR.

An abbreviated notation is F f = f * dF.

Lemma.

i) Fn ---+ Fo in the strong operator topology iff Fn ---+ Fo in distri

bution. That is) for Fn E 1):.F, IIFnf - Fofll ---+ 0 for all f E X

iff Fn (x) ---+ Fo (x) for all x E JR at which Fo is continuous.

ii) (F * GF = FG.

The elementary proof is omitted. Note that i) says that

F ---+ F is bicontinuous; ii) says that it converts convolution

to ordinary multiplication (or composition) of linear operators.


This is significant since for independent random variables the dis

tribution function of the sum is the convolution of the distribution

functions.

A sequence e = {el, 6, ... , en, ... ,} of random variables is

called independent if for every finite subset {nl' ... , nk} of IN and k

for all Ak E 'E, P{eni E Ai for 1 ::; i::; k} = n P{eni E Ad· i=l

Equivalent is that F k = FEnl *FEn2 * .. . *FEnk · The sequence I: Eni i=l e is called identically distributed if F = FEn is independent of n.

CENTRAL LIMIT THEOREM.

Let ei, e2, . .. be independent identically distributed random

variables with mean 0 and variance 1. Then as N -+ 00, the N

normalized sum J& I: ei converges in distribution to a standard ,=1

normal random variable.

Some of these terms must be explained. For Borel functions

g on JR,

with the understanding that whenever one integral exists, they

both do and equality holds. The most important numbers of this

form are the mean

and the variance

Here we drop the subscript i from F since the ei are identically

distributed; the two functions used here are g(x) == x and g( x) ==

26 1. A. Goldstein

(x - J-l? If e has a finite variance, i.e. if e E L2(n, E, P), then

replacing ei by 'fJi = ei;JL gives that 'fJi has mean 0 and variance

1. This holds without loss of generality for nonconstant random

variables (i.e. (J' > 0). The distribution function for the standard

normal distribution is

Thus the conclusion of the Central Limit theorem is that for all

x E JR,

1 jX 2 lim F N (x) = ro= e-t /2dt. N-oo _1_ L: e. v27r- 00

VN i=l z

N Note that 'fJ = J-& ?= ei is the normalized sum of 6,· .. ,eN in

z=l that 'fJ has mean 0 and variance 1.

Here is a quick outline of the proof.

Let X = BUC(JR),

D = {j E X n C2(JR): j',j" EX},

G = Fei the common distribution function and for r > 0 let

Gr(x) = G(vrx) for x E JR. If V(t) = GIlt with V(O) = I, then

the hypotheses of Chernoff's theorem hold. A Taylor's theorem

argument gives V'(O)j = ~j" for JED. Chernoff's formula

implies

V(!tj~T(t)j. n

But V (;) = Fq with 'fJ = If t ei; and SInce A = i=l

~d2/dx2, T(t) = FN(o,t), i.e., the semi group governing the heat

equation ( with the factor 1/2) is given by convolution with the


distribution function of the normal distribution with mean 0 and

variance t. The result follows with the aid of the lemma.

LIE-TROTTER-DALETSKII PRODUCT FORMULA.

Let A, A2 and A3 = Al + A2 be densely defined m-dissipative

operators on X. Let Ti be the semigroup generated by Ai. Then

[ t t ] n T3(t)f= lim TI(-)T2(-) f

n-+CX) n n

holds for all t ~ 0 and all f E X.

This follows from Chernoff's formula. Take V (t) = TI (t )T2 ( t)

and calculate V'(O) = Al + A2 on V(Ad n V(A2)'

FEYNMAN PATH FORMULA.

The wave function of a (nonrelativistic) quantum mechanical

system in the solution u = u( x, t) of the Schrodinger equation

i OU = __ 1_6u + V(x)u (x E IRi , t E IR). ot 2m

u(x,O) = f(x) (x E JRE).

Imagine N identical particles (say electrons), each of mass m.

Then £ = 3N and the real function V is the potential describing

the forces on the particles. It is assumed that the closure H

of ~~ + V ( on V(6) n V(Mv)) is selfadjoint; this is a very

mild restriction on V. By Stone's theorem, -iH generates a (Co)

unitary group U, and u(t) = U(t)f for t real. The datum f is

normalized to be a unit vector in L2(IRi); then Ilu(t)112 = 1 for all

real t. lu(x, tW [resp. lu(e, tW] is the position [resp. momentum]

probability density of the system. That is, fr lu(x, tWdx is the

probability that at time t the position vector of the system is

28 J. A. Goldstein

in r c ]Rf. Both lu(·, t)12 and lu(·, t)12 are probability densities

because of the unitarity of the Fourier transform on L 2(]Rf).

Apply the Lie-Trotter-Daletskii product formula with A = iD./2m, B = M-iV and C = A + B = -iH. We have

(TA(t)f) (x) = (27rit/m)-1/2 [ eimlx-yI2/2tf(y)dy, Jm}

(TB(t)f) (x) = exp {-itV(x)}f(x).

If Sn(t) = (TA(t/n)TB(t/n))n, then by induction,

n (7)

where

Define the space nx of continuous paths starting at x:

nx = {w E C(]R+, ]Rf) : w(O) = x}.

If Xj = w(jt/n) and Xo = x, then S is a Riemann sum approxi

mating the action integral

S(Wit) = it [;lw(sW - V(w(s))] ds.

Formally

(Sn(t)f)(x) 4 C [ eiS(w;t) f(w(t))Vw. (8) Jflx

This is the celebrated Feynman integral formula for the wave fun

tion u(x, t). There are at least three mysterious aspects of it.

First, the "constant" C satisfies ICI = lim 1 211" it l-f / 2 = 00. Next, n-+oo nm

Seroigroups of Linear Operators 29

the "measure" Vw = IT dxs (= lim dXl." dXn) cannot be O<s<t n-+oo

defined as a count ably additive set function. Kac considered the

associated heat equation u' + H u = 0, u(O) = f and represented

the solution in a similar way as a Wiener integral over nx ' For

Wiener integrals, paths w which are somewhere differentiable form

a null set. In other words, if one could interpret (8) as a Wiener

intergral, then the integrand involves Sew; t), which for t > 0 is

infinite almost everywhere. Thus in the mysterious Feynman for

mula (8) neither the constant nor the measure nor the integrand

makes sense. But the formula

u(X, t) = C r e i S(w,t) f(w(t))Vw Jn x (9)

has great insuitive appeal. It expresses a basic object of quantum

mechanics as an average over paths involving classical mechan

ics notions, namely, the action. Feynman used this idea to great

profit in his Nobel Prize winning work on quantum electrodynam

ics and elsewhere.

It is now easy to give a ngonous interpretation of (9).

Namely, replace (9) by the Riemann like approximant (7). Then

by the Lie-Trotter- Daletskii product formula, (8) holds, in the

sense that Sn(t)f converges in L2 (JRl') to the solution u(·, t).

The beautiful argument is due (independently) to E. Nelson

and Y. Daletskii.

INHOMOGENEOUS EQUATIONS.

Consider

u' = Au + h, u(O) = f

where A generates a (Co) semi group T on X. The variation of

30 J. A. Goldstein

parameters formula represents the solution (if it exists) as

u(t) = T(t)! + it T(t - s)h(s)ds.

If one regards T( t) as et A, this is exactly the result of one variable

calculus for the ODE u' = au + h(t). The moral is that you can

make a living in this subject if you are really proficient in one

variable calculus.

The semilinear problem

u'(t) = Au(t) + get, u(t)), u(O) = !

can be attacked by successive approximations of the form

The solution of this linear inhomogeneous problem is, by the

above, given by

un(t) = T(t)! + it T(t - s)g(S,U n-l(s))ds.

Thus solving our problem reduces to solving the integral equation

u(t) = T(t)! + it T(t - s)g(s, u (s))ds.

If (Su)( t) is the right hand side of the above equation, then one

wishes to find a fixed point of S, viewed as a mapping from a

subset of C([O,rJ,X) into itself. Such a fixed point is called a

mild solution of u' = Au + g( t, u), u(O) = f. It may not be differ

entiable (depending on g), but if it is it is a solution in the usual

sense. Also, in showing that S has a fixed point, one may have to

take r be be small (and positive), thus one gets a local solution.

An additional a priori estimate is needed to get global existence,

which does not always hold.


THE NAVIER-STOKES EQUATIONS.

Let u(t,x) E IR? be the fluid velocity and p(t,x) E IR the

pressure of a fluid occupying a ( smooth bounded) container n c IR3. Here x E nand t ;:::: 0 is the time. The motion of the

(incompressible) fluid is given by the N avier-Stokes equations,

Ut - vfl.u = -(u· V)u - Vp + 9 in[O, T] x n,

div u = 0

(NS)o

u=O

u = Uo

This can be rewritten as 3

au' L au' ap _J _ vfl.u· = - Uk-J __ + g' at J ax ax' J,

h=l

k=l k J

u (t, x) = 0 (t ~ 0, x E an),

u(O,x)=uo(x) (xEn).

Define the spaces

'H == L2(n; (C3) = [L2(n)r,

in[O, T] x n,

on[O, T] x an,

on{O} x n.

j = 1,2,3,

'Hu == cC1l {uc1(n; (C3): div u = 0 in n, u = 0 on an},

'Hv == cC1l {V'P : 'P E c1(nn· Here cC1l means the closure is 'H and u stands for solenoidal (or

divergence free) vectors.

Lemma. 'H = 'Hu EB 'Hv.

32 1. A. Goldstein

This observation, which goes back to Helmholtz, is often called

the Hodge decomposition. To prove the orthogonality, note that

(u, "Vcp) = 1 U· "Vcpdx = 1 { div (cpu) - cp div u}dx

= r cpu. ndS = 0 Jan by the divergence theorem where n is the unit outer normal and

the fact that div u = O. It remains to write any u E [c,:x'(n)t as u = v + "Vcp where div v = O. Let h = div u. From div u = div v+ div "Vcp, it follows that fj.cp = hand ocp/on = "Vcp·n = u·

n = 0 on an. Solving this Neumann problem gives cp; v = u - "V cp

gives the lemma. o

Let P : 1-i -t 1-iu be the orthogonal projection onto 1-iu' The

Stokes operator is

A=VPM,X3=VP(! * 1) where fj. is the Dirichlet Laplacian: D(fj.) = H2(n) n HJ(n).

The Stokes operator is morally like the Dirichlet Laplacian.

Lemma. A = A * S -d on 1-i for some E > O. A has an or

thonormal basis of eigenvectors.

Let v = Pu (which is u if u is 1-iu valued). Then (N S)o

becomes

{VI (t) = Av(t) + N(v(t)) + h (t),

v(O) = Vo.

Here Vo = PUo = Uo, h = Pg, N( v) = -P( v . "V)v. The point is

that P("Vp) = O. Thus (NS)l is an equation in 1-iu. After solving

(N 5h uniquely for v, set u = v and plug into (N 5)0. This gives

\1 p, which determines p uniquely up to an additive constant.

Following earlier work of Leray, Hopf, and others, Kato and

Fujita established local wellposedness of (N 5h; i.e. they found

a solution for t E [0, T] for some T > O. Here is a outline of the

argument.

Let K be a Hilbert space and Ao a selfadjoint operator on

K satisfying Ao = A~ :::; -d. (Think of K = Ho and Ao = A.)

By the spectral theorem, Ao is (in some representation) multipli

cation by a function with values in (-00, -c:] on some L2 space.

So define Ka = V(( -At) with norm Ilflla = II( -Ao)a fll. Let

o :::; a < 1 and let F map an open set containing f in Ka to K

and satisfy

IlFh - Fhll:::; I<llh - hila

for some I< = I< f and all fl' h in the open set. Then there

is a unique mild solution of v' = Av + F( v), v(O) = f in

C([O,T],X) for some T > O. Since xQe-Q :::; 1 for x > 0, the

spectral theorem implies 11(-Ao)aetAoll:::; ra. Hence

lilt e(t-s)AO(F(Vl(S)) - F(V2(S)))dsll

:::; lt Ile(t-s) AOII B(lC,lC Q )IIF(VI(S)) - F(V2(S))liK Q ds

:::; I< It(t - s)-Qllvl(S) - V2(s)II Q ds,

and the integral exists for t > 0 since (t - s) -Q is integrable on

[0, t] because a < 1. The Banach fixed point theorem (alias the

Picard iteration method) gives the desired local solution.

34 J. A. Goldstein

To apply this result to (N 5)1, one must use Sobolev inequal

ities; the end result is that any ()' E (3/4,1) will suffice.

The problem of global wellposedness of (N 5)1 is still an open

problem. Its solution will explain turbulence.

A SEMILINEAR PROBLEM WITH A NONLINEAR

BOUNDARY CONDITION.

Here we give another example of solving a nonlinear problem

by solving linear inhomogeneous problems.

Let X = LP(D) where D is a smooth bounded domain in IRn.

Let A be the Laplacian ~ with domain V(A) = W 2 ,P(D). We

have W 2 'P(D) '---+ C(fl) provided max{~, I} < p < 00, which we

assume. Let Bu be the restriction of u to oD. Think of B : X -+ Y,

where Y is a space of functions on the boundary oD. We are

interested in cases where B is A-closed, i.e. in E V(A), in -+

i, Ain -+ g, Bin -+ h imply i E V(A) and Bi = h. In our

application, B is not closed but it is A-closed.

Of concern is the problem (with general A, B)

u'(t) = Au(t) (t ~ 0),

u(O) = i,

Bu(t) = 9 (t ~ 0).

The compatibility condition at t = 0 is

Bi =g.

Express the solution as u(t) = U(t;i,g).

(10)

We assume that this problem has a unique solution u and

the function U satisfies

IIU(t;i,g)ll::; Mewt(llill + Ilgll)


for some M 2:: 1, wEIR and all t,j, g such that (10) holds.

Now consider

u'(t) = Au(t) + F(t) (t 2:: 0),

u(O) = j,

Bu(t) = w(t) (t 2:: 0).

35

We assume that A, the restriction of A to the null space IN(B)

of B, is the generator of a ( Co) semi group T. (In our concrete

example, A = ~ defined on V(A) = W2 ,p(n) n W~,p(n).)

Formally let

where

U1(t) = T(t)j + it T(t - s)F(s)ds,

U2(t) = U(t;O,w(O) + it U(t - s,O,w'(s))ds.

Calculating gives

U{ = AU1 + F, U1(0) = j, BU1 = 0,

U~ = AU2, U2(0) = 0, BU2 = w.

Unfortunately this is not quite right because U( t; j, g) is assumed

to exist and have nice properties only when B j = g.

We get around this by solving the abstract Dirichlet problem

Au = 0, Bu = g.

In our concrete example, this is ~u = 0 in n, u = g on an. Call the solution u = Gg. Thus we assume there is aGE B(Y, X)

such that AGg = 0, BGg = g for all g E Y.

36 1. A. Goldstein

Now replace A by A - AI for suitable real A. Then

v = A1 00 e->'tU(t;f,g)dt - A(AI - A)-l j

satisfies (AI - A)v = 0, Bv = g. Thus our incorrect formula

u = U1 +U2 for the solution can be replaced by the correct formula

u(t) = T(t)(f - Gw(O)) + U(t; Gw(O), w(O))

+ 1tT(t-S)(F(S)-Gw'(S))dS

+ 1t U(t - S; Gw'(s), w'(s))ds.

One can verify that u is a solution in C1(ffi+, X) provided

w E C1 (ffi+ , Y), j - Gw(O) E D(A), and F - Gw' E

C1(ffi+, X) + C(ffi+, Z) where Z = D(A), equipped with the

graph norm. The formula for u is a kind of variation of parameters

formula (or Duhamel formula) involving inhomogeneous "forcing

terms" and inhomogeneous boundary conditions.

This leads to the solution of semilinear problems in the ex

pected way. So consider

u'(t) = Au(t) + F(t, u(t)),

u(O) = j,

Bu(t) = N(u(t)),

where F and N are nonlinear but locally Lipschitzian (on suitable

spaces). The successive approximation approach gives

where

u~(t) = Aun(t) + hn(t),

un(O) = j,

Bun(t) = wn(t),


Solve this and use the Banach fixed point theorem to get a local

(in time) solution of the semilinear problem. Global existence

follows in situations where one can establish an a priori bound.

MEAN ERGODIC THEOREM.

Let A generate a (Co) contraction semigroup T on a Hilbert

space H. Then

liT lim - T(t)Idt = Pof. T-+(x) T 0

Here Po is the orthogonal projections of H onto the null space

of A, IN(A) = lR(A)-L. For the proof, let F = fI + fz E IN(A) + lR(A). Then AfI = 0, so T(·)fI and fI are both solutions of

u' = Au, u(O) = II; uniqueness implies T(t)Il = II for all

t ~ O. Next,

T- 1 iT T(t)fzdt = T- 1 iT T(t)Agdt =

TIlT dd (T(t)g)dt = T(T)g - 9 = 0(1) otT

as T --t 00. It follows that

lIT - T(t)(fI + fz)dt --t fI T 0

for fI + fz E IN(A) + lR(A). In particular,

IN(A) n lR(A) = {O}.

In the above arguments T and A can be replaced by T* = {T(t)* :

t ~ O} and A* respectively. Thus IN(A*) n lR(A*) = {O}. But

H = {o}-L = (IN(A*) n lR(A*))-L = lR(A) ED IN(A).

38 J. A. Goldstein

Since II~ foT T(t)fdtll ~ Ilfll, thus IIPII ~ 1 and the result follows.

o

Let 1-lS(1-l) be the set of all Hilbert-Schmidt operators on

1-l. For B a compact operator on 1-l, let IBI = (B* B)1/2; by the

spectral theorem, 00

IBI = LAn < ., en > en n=l

where {en : n E N} is an orthonormal basis for 1-l. (Here we are

taking 1-l to be separable, although this is really not necessary.)

The Hilbert-Schmidt norm of B is defined by 00

IIIBIW = II{An}II~2 = L A~, n=l

and the inner product on the Hilbert space 1-lS(1-l) is given by

Let A be as in the mean ergodic theorem. Define U = {U(t) :

t E ]R+} by

U(t)B = T(t)* BT(t),

for BE 1-lS(1-l). Then U is a (Co) contraction semigroup, and its

generator G is given by

G(B)=A*B+BA;

G(B) = [B, A] holds when T is a unitary group. For fj a unit

vector in fi, let B j = (-, Ii) Ii = Ii 0Ii. Then

as a short calculation shows. The mean ergodic theorem implies


and G(B) = 0 for B E HS(H) implies B is a linear combina-

tion (or infinite series) of finite rank projections corresponding to

purely imaginary eigenvalues of A. Thus we deduce

1 fT -:;: io I(T(t)fl,hWdt -+ L I(PAiI,hW· o AEup(A)niIR

This is Wiener's theorem. (Intuitively for the unitary part of

T, G(B) = 0 = AB - BA, and the only compact operators which

commute with A* = -A are built from rank one projections onto

eigenvectors of A. For the nonunitary part of T, I(T(t)fl,h)1 2

formally tends to zero as t -+ 00.)

For a specific example, let P be a finite Borel measure on lR,

and let H = L2(lR, p). Let A be multiplication by i times the

identity function on H. Then (T(t)f)(x) = eitx f(x) for f E H, x

and t E lR; and

(T(t)l, l) = i: eitxp(dx) = j1(t),

is the (nonnormalized) Fourier transform of p; and

- 1j1(tWdt -+ f liT T 0

by Wiener's theorem, where f = E{ (PAl, 1) F. Unraveling what

this means yields that f is the sum of the squares of the jumps

associated with p. That is, if p = Pc + Pd is the Lebesgue de

composition of p and if the discrete part is given by Pd = Ea j 8x ., J

then f = Ea~.

Now let L be a bounded linear operator on H such that

L(>.o - A)-l is compact for some>. in the resolvent set of A.

Then Wiener's theorem can be used to prove

40 1. A. Goldstein

! iT IILT(t)fWdt -+ L IILP,\fI12 • (11) T ° '\Eo-p(A)niIR

This has the following interpretation when A = -i(-~ + V(x))

and L = MXB(O,R) ' i.e. iA is a selfadjoint Schrodinger oper

ator on V(A) = H2(JR) c 1i = L2(JR1) and L is multiplica

tion by the characteristic function of the ball of radius R. (See

the earlier section on the Feynman path formula.) Let u satisfy

Ut = Au, u(O) = f where f is a unit vector in 1i. Then lu(x, t)12 is

the position probability density of the system, i.e. fr lu(x, t)1 2dx

is the probability that the position vector of the system lies in

r c JRl at time t. We have

lit liT] - IILT(t)fWdt = - lu(x, t)1 2dxdt, TOT ° Ixl<R

and

lim lim! iT] lu(x, tWdxdt = 1 [resp. = 0] (12) R--+oo T--+OO T ° Ixl<R

for bound states [resp. for scattered states, or for states that are

asympotically free, and leave each bounded set as time gets large].

By (11), the above limit is

lim '" IIP,\fW = 1 [resp. = 0] (13) R--+oo ~

'\Eo-p(A)

when f is in the closed span 1id(iA) of eigenvectors of A [resp.

when f is orthogonal to this space]. For H = H* = f~oo >.dE(>.)

a self adjoint operator on a Hilbert space 1i, we say that f is in

the subspace 1iac( H) of absolute continuity of 1i iff the monotone

function>. -+ IIE(>')fW is absolutely continuous on JR. Similarly

for 1isc(H), 1ic(H), 1id(H), where sc [resp. c; resp. d] means sin

gular continuous [resp. continuous; resp. discrete]. A monotone


function is discrete when it is constant except for jumps. The

Lebsegue decomposition theorem for selfadjoint operators says

Hc(H) = Hac(H) EB Hsc(H),

H = Hc(H) EB Hd(H),

Hd(H) = span { all eigenvectors of H}.

What was shown in (12) and (13) together with the Lebesgue

decomposition theorem implies that the scattering states [resp.

bound states] are the vectors in Hc(H) [resp. Hd(H)] for H = iA.

In many applications, Hsc(H) = {O}, and the scattering states are

POSITIVITY AND SECOND ORDER OPERATORS.

If A is a (weakly) elliptic (partial) differential operator of order

two, then by the maximum principle, the semi group T generated

by A is positivity preserving. So is the translation semigroup

generated by the first derivative. A sort of converse holds; we

indicate this is only the simplest situation.

Consider the lattice of real functions L2(n, f.1.) Suppose a

(Co) semigroup T is given by

(T(t)f)(x) = in K(t,x,y)f(Y)f.1(dy)

for all t, f, x. Then K ~ 0 a.e. on IR+ x n x nand f ~ 0 implies

T(t)f ~ 0, i.e. T is a positivity preserving semigroup. From

(,\ - A)-l f = /00 e-AtT(t)fdt,

T(t)f = lim (I - !A)-n f, n ...... oo n

it follows that T( t) is positivity preserving for all t ~ 0 iff ().-A)-l

is positivity preserving for large). > O. But we want to know

42 1. A. Goldstein

criteria for this involving A directly, not (A - A)-I.

Let A = A * be a nonpositive selfadjoint operator on 1i =

L2(n, J.L), a real Hilbert space which we view as a Banach lattice

of real functions. Let the (Co) semigroup T generated by A be

positive or positivity preserving in the sense that f E 1i, f ~ 0

implies T(t)f ~ O. Then, since f = f+ - f-,

and consequently , by intergrating over n,

g(t) == IIT(t)fI12 :::; IIT(t)f+112 + IIT(t)f-W == h(t).

For f E V(A), from 9 :::; h on [O,ooJ and g(O) = h(O) we deduce

g'(O) :::; h'(O), i.e.,

Hence

! (T(t)f, T(t)f}lt=o = (Af'!) = -II( _A)1/2 fW

:::; -11(-A)1/2f+W -11(-A)1/2f_W.

and so f E V(A) [or f E V((-A)1/2)J implies f+,f- E

V(( _A)I/2), and the same holds for If I· Suppose A is an elliptic

operator of order 2m, so that

where these are the usual Sobolev spaces. Then V(( _A)1/2) C

Hm(n), and so f E H~m(n) implies If I E Hm(n). This implies

m :::; 1. The simplest way to see this is to think about f( x) ==

Seruigroups of Linear Operators 43

x near x = 0; f E Hm( -1,1) for all m, but If I E Hm( -1,1)

holds only for m :s: 1.

SCATTERING THEORY.

For j = 0,1, let Hj be a selfadjoint operator on a Hilbert

space 'H and let Uj be the corresponding (Co) unitary group gener

ated by -iHj ; thus Uj(t) = e-iHj governs the Schrodinger equa

tion idu/ dt = Hju. The free equation [resp. perturbed equation]

corresponds to j = 0 [resp. j = 1]. V = HI - Ho is the perturba

tion. Think of V as a scattering source; thus Ul (t) = UI (t)f looks

like a free solution for t large and negative and Ul(t) looks like

a different free solution for t large and positive. More precisely

there are vectors f ± E 'H such that

lim I lUI (t)f - Uo(t)f±1I = O. t----±oo

Thus f = lV±f± where

are the wave operators. Here Po is the orthogonal projection

onto 'Hac(Ho); when TV±(H1 , Ho) exists, it is an isometry from

'Hac(Ho) into 'Hac(Hl)' W± are called asymptotically complete

when they exist and their ranges are both 'Hac(HI)' The scattering

operator S maps f _ to f +. Thus

f+ = W;lf = W;IW_f_

and so S = W;IW_ is unitary on 'HacCHO) when W± ex

ist and are complete. By the chain rule, tV±(H2,Ho) =

W±(H2' HdW±(Hl, Ho), and so existence and completeness of

W±(HI, Ho) follows from existence of both W±(Hl' Ho) and

W±(Ho,Hd·

44 1. A. Goldstein

A closed densely defined operator A is said to be H -smooth

provided there is a constant C such that i: IIAU(t)/11 2 dt ~ C2 11/W

holds for all 1 in a dense set in 1i, where H is selfadjoint and

U(t) = e-itH . This implies

IR(A*) c 1iac(H).

Proof: Write H = J~oo '\dE('\). For 1 E V(A*),

a(A) == (E('\)A* I, A* I) = IIAE(A)gW

(where 9 = A * f) is bounded and nondecreasing, and

.)2;u(O = i: e-ie>'d(E(A)g,g)

= (U(Og,g) = (AU(Og,f);

271" i: lu(eWde ~ 11/112 i: IIAU(OgWde < 00

since A is H-smooth. Thus u E L2(IR) and so the monotone

function a is absolutely continuous and its derivative is in L2(IR),

by uniqueness of Fourier transforms. Hence 9 E 1iac( H). 0

The notion of (Kato) smoothness can be localized. For r a

Borel set in IR, A is H-smooth on r means AE(r) is H-smooth

where

E(r) = l dE('\).

THEOREM.

Let HI = Ho + Ai'Ao in the quadratic form sense [i.e.

V(Aj) ~ V(Hj), (HI/,g) = (j,Hog) + (AI/,Aog) for 1 E


1)(H1), 9 E 1)(Ho) j. Suppose Aj zs Hj-smooth on r for

j = 0,1. Then

W±(r)j = lim U1( -t)Uo(t)EO(r)j, t--+±oo

n±(r)j = lim Uo( -t)U1(t)EI(r)j t--+±oo

exist for all j E H, where Hj = I~oo >.dEj(>.), Ej(r)

IrdEj(>'), j = 0,1. Ifrn = (an,bn) C JR is a sequence of

intervals with u(Hj)\ U~=I r n at most countable, and if, for

j = 0,1, Aj is Hj"smooth on r n for each n, then the wave

operators W ± ( HI, H 0) exist and are asymptotically complete.

This can be applied to Ho = -~, HI = -~ + Mv on

L2(JRn), where V is real and belongs to LP(JRn) + LOO(JRn) for

some p > n/2 (with p > 1), and V is a short range potential,

i.e., W(x)1 ::; constant Ixl-I- e as Ixl --+ 00. The idea is to take

Aj = MWj where Wo = IVI I / 2 , WI = ( sign V)IVI 1/ 2 • One ap

peals to Weyl's theorem that, since HI is a relatively compact

perturbation of Ho, uess(Hd = uess(Ho) = [0,00). Also any

eigenvalues of HI in (0,00) are isolated. Thus the r n are chosen so

that each interval r n is in JR+, is a positive distance from up(Hd,

and unrn = (O,oo)\Up(HI). This result was proved by Kato and

Kuroda (using different methods.) The notion of smoothness is

due to Kato. Verifying that potential perturbations of elliptic

operators are smooth is usually based on work of Agmon. Exten

sions of the Kato-Kuroda theorem from two body problems to N

body problems was done by Enss, Sigal, Soffer and others. The

books of Reed and Simon give valuable additional information.

We indicate the proof the above italicized thorem in the case

when r = JR.

46 1. A. Goldstein

Let w(t) = U1( -t)Uo(t)h. It suffices to show lim w(t) ex-t-++oo

ists. The same argument applies to lim w(t) and for w(t) = t-+-oo

Uo(-t)U1(t)h since the hypotheses are symmetric in the indices

0,1. Completeness of the wave operators then follows from the

chain rule. Suppose h E V(Ho), k E V(HI). Then for t > s,

~ (k, w(t)} = -i{ (HI U1 (t)k, Uo(t)h)

- (U1(t)k, HoUo(t)h)}

= -i(Al U1 (t)k, AoUo(t)h),

I(k, w(t) - w(s )}I ::; it I(AU1( r)k, AoUo( r)h}ldr

S ([ IIAUI(r)kll'dr) 1/'

. ([ IIAoUo( r )hll'dt) 1/'

= J1 + hi

J1 ::; Gllkll and J2 --+ 0 as s, t --+ 00, since Aj is Hj-smooth for

j = 0,1. Taking the supremum over Ilkll ::; 1 gives that w(t) is

Cauchy as t --+ 00. 0

EQUIPARTITION OF ENERGY.

Let A be an injective selfadjoint operator on a Hilbert space

1i. Of concern is the abstract wave equation

u"(t) + Au(t) = 0 (t E lR),

u(O) = iI, u'(O) = h.

Define the kinetic energy, potential energy, and total energy at

time t to be

K(t) = Ilu'(t)W, P(t) = IIAu(t)W, E = K + P.


Think of K ( t) as the sum ( or integral) over the space variable

of the square of the velocity (times the constant density over

two); thus K( t) is the usual kinetic energy for Utt = c2uxx for

x in an interval J (say with Dirichlet boundary conditions) and

K(t) = IJ IUt(t, x Wdx.

The total energy is conserved. This follows from

d E'(t) = dt {(u'(t), u'(t)) + (Au(t), Au(t))}

= 2Re{ (u'(t), u"(t)) + (u'(t), A2u(t))} == 0

since A* = A and since u" + A2u = O. By the spectral theorem,

we can write the unique solution of the abstract wave equation as

u(t) = cos(tA)JI + A-I sin (tA)h.

(Note that for A selfadjoint, A -1 sin tA is a well-defined bounded

linear operator, even when A fails to be injective, since (sin tx) /

x --t t as x --t O. Since

2u'(t) = eiL4 g1 + eitAg2'

2iAu(t) = eitAg1 - e-itA g2,

with g1 = iAJI + h, g2 = -iAJI + h, it follows that

K(t) = lIu'(t)W = Ilgl W + IIg2W + 2Re(e2itAgl,g2),

P(t) = IIAu(t)W = IIg1W + IIg2W - 2Re(e2itAgI,g2).

Since ]{ + P = E, and by the parrallelogram law,

we conclude that as t --t ±oo,

K(t),P(t) --t E/2 (14)

for all solutions if and only if eitA --t 0 in the weak operator

topology iff (for A = J::oo )"dE()"))

48 1. A. Goldstein

(15)

for all h E H. (In the last condition, (E(A)h, h) can be replaced

by (E(A )h, l) using the polarization identity.) The condition (14)

is called asympotic equipartition of energy.

A selfadjoint operator A satisfying (15) is called a Riemann

Lebesgue operator. More generally, for A = J~oo AdE( A) selfad

joint on H, let HRL(A) be the set of all vectors h E H for which

(15) holds. Then it can be shown that HRL(A) is a closed sub

space of Hand

and each containment can be strict here. Since ~ on L2 (lR n)

( and (_~)1/2 as well) is spectrally absolutely continuous, it fol

lows that the concrete wave equation

Utt=~U (x ElRn , tElR)

admits asymptotic equipartition of energy.

An extension of higher order problems was obtained by Gold

stein and Sandefur. We shall indicate this in the next section.

ABSTRACT d'ALEMBERT FORMULA.

Let Aj generate a (Co) semigroup Tj = {Tj(t) : t E lR+}

on X for j = 1, ... , N. We assume that the Tj are mutually

commuting families:

holds for all j, k, t, s. Of concern is the equation

J1(~-Aj)U(t)=O (16)


We say that d 'Alembert 's formula holds if all solutions are of the

form

N

u(t) = LTj(t)/j. j=l

The classical example is

Utt - u = (~ - ~) (~ + ~) u = 0 xx dt dx dt dx

with X = L2(lR). Then every solution (in L2(lR)) is given by

u(t,x) = fl(X + t) + f2(X - t)

= (et(+d/dX) It) (x)

+ (et(-d/dX)h) (x).

Before generalizing this substantially, we note first that it does

not always hold. In case A1g = A 2g for some vector 9 =1= 0, then

is a solution of (16) with N = 2.

We shall show informally why d'Alembert's formula holds

whenever Aj - Ak is injective and has sufficiently large range for

j =1= k. (The above example Al = -A2 = d/dx on L2(lR) indicates

that we do not want to assume that Aj - Ak is surjective.) We

give the argument for N = 2 for convenience.

So consider (d/dt-Adv(t) = 0 where v(t) = (d/dt-A2)u(t).

Then by the variation of parameters formula,

50 J. A. Goldstein

u(t) = T2(t)U(O) + 1t T2(t - s)v(s)ds

= T2(t)U(O) + 1t T2(t - S)TI(S)V(O)ds

i t d = T2(t)U(O) + 0 d)T2(t - s)TI(v)g]ds

= T,(t)[u(O) - gJ + T,(t)g ( = ~ T;(t)f.)

provided 9 = (AI - A2)-I v(O). Thus v(O) = u'(O) - A2U(O)

should belong to the range of Al - A2 •

The abstract wave equation u" + A2u = 0 (with A = A* on

1{) can be written in factored from as

(d/dt - iA)(d/dt + iA)u(t) = O.

Relative to v = (~:), this becomes an abstract Schrodinger

equation on 1{2 of the form

Vt = Av,

Let Pj be the orthogonal projection onto the jth component in

1{2. Then

and as noted before, K(t),P(t) -+ E/2 for all initial data (see

(14)) if and only if A is Riemann-Lebesgue operator on 1{ (in

symbols, A E lRJL(1{)), i.e. (15) holds, or 1{RL(A) = 1{.

This formulation of equipartition of energy leads to a signif

icantly deeper result due to Goldstein and Sandefur. Consider n

II (d/dt - iAj)u(t) = 0 j=l


where AI, ... ,An are commuting selfadjoint operators on 1i. The

commuting hypothesis means that exp( it Aj) and exp( is Ak )

commute for all t, s, j, k. When n = 2N is a power of 2, this

equation can be written in the form

v' = iAv.

where A is a selfadjoint operator on IC = 1in. In this particular

way of writing our factored equation as a system, let Pj be the

orthogonal projection onto the jth component of IC; thus IC = n

EB PjIC. Let Ej(t) = IIPjv(t)W be the jth partial energy of a j=I

solution u. Then the theorem is:

n E = L: Ej(t) is independent of t; and Ej(t) --+ E/2N as t --+

j=I

±oo for each j = 1, ... ,2N and all solutions if and only if Aj -

Ak E IRIL(1i) whenever j =I- k.

The equations of linearized elasticity can be written in this

form. Let u be any component of the shear wave or pressure wave.

Then u satisfies n

2:)d2 /de - AjL:~)u(t) = 0 j=l

where ~ is the Laplacian on 1i = L2(IR3 ) and AI, A2 are the

(positive) Lame parameters. This can be written as a factored

equation of order 4, and equipartition of energy holds if and only

if Al =I- A2.

Now consider the telegraph equation (or damped wave equa

tion)

u" + bu' + A2 u = 0, (17)

52 1. A. Goldstein

where A = A* on 'H and b is a positive constant. If K(t) = Ilu'(t)W, P(t) = IIAu(t)W as before, then K(t), P(t) ~ 0

as t ~ 00, but one can still have equipartition of energy if K

and P tend to zero at the same rate. The d'Alembert formula

leads to the following result. If A is absolutely continuous (i.e.

1i = 'Hac(A)), A2 ~ a2 I, and if 0 < b < a/2, then for all nonzero

solutions of (17),

K(t) -- ~ 1 as t ~ 00. P(t)

SCATTERING REVISTED.

Consider the factored equation

for k = 0,1 and t E JR. Here Hy) is a selfadjoint operator on 1i.

If UY\t) = exp( -itHY\ then the d'Alembert formula says that

in many circumstances the solutions of (17)k are all given by

n

u(t) = L UY)(t)fj. j=l

Suppose that Kk = 'Hac (HY») is independent of j. Suppose

that, for each j, the wave operators

exist and are asymptotically complete. This enables one to show

that for each solution v of (17h, there are solutions u± of (17)0

such that

Ilv(t) - u±(t)11 ~ 0

as t -+ 00. Using this one can construct wave and scattering oper

ators for (17)k. With this observation, together with the Birman

Kato invariance principle, one can do scattering theory for wave

equations with potentials and for elasticity equations with poten

tials as a consequence of the known potential scattering theory

for Schrodinger operators.

REMARKS ON NON (Co) SEMIGROUPS.

The solution of the heat equation on lRn is given by

u(t,x) = (T(t)f)(x) = (47rt)-n/2 J e-lx-YI2/4tf(y)dy. lRn

T defined in this way defines a (Co) semI group on vanous

spaces, e.g., LP(lRn ) (1::; p < 00), Co(lRn), C(lRn ), and

BUC(lRn). But this formula actually defines a semlgroup on

larger spaces, for instance X = L<Xl(lRn). But T on X is not

strongly continuous at t = O. It satisfies T(t) : X -+ BUC(lRn)

for t > 0, and the largest subspace of X on which T is strongly

continuous on lR+ is BUC(JRn ).

Now consider

Ut = 6u + h(t), u(O) = f

where 6 is the Dirichlet Laplacian acting on real or complex func

tions on n, a smooth bounded set in lRn. 6 generates a (Co)

semigroup on Y = Co(n), but it determines a semi group T on

X = C(Q), which is not strongly continuous at t = o. The varia

tion of parameters formula gives

u(t) = T(t)f + it T(t - s)h(s)ds. (18)

54 J. A. Goldstein

This is classical for h : IR+ --t Y, but in this case h( t, x) should

vanish for XEOn. This is not always reasonable, and one wants

often to use (18) for h : JR+ --t X.

Here is another example of a non (Co) semigroup. Let X =

Co[O, 1) = {f E C[O, 1] : f(l) = O}. One can view X as a subspace

of Co(JR+) by defining f(x) = 0 for x > 1 and f E X. Let Al =

d/ dx and A2 be multiplication by V E Ll [0,1]. The Daletskii

Lie-Trotter product formula gives that the semigroup T generated

by A = Al + A2 is given by

(T(t)f)(x) =exp{l t V(x+s)ds}f(x+t),

smce

[ t t]n n t

Tl ( - ) T2 ( - ) f ( x) = exp {L -V (x + j t / n )} f (x + t). n n . n

)=1

If we take V(x) = c/x, so that

t ( + t) (+ t)C exp{}o V(x + s)ds} = exp{clog =-;- } = =-;- , then we get a semigroup T given by

T(t)f(x) = (x: t) C f(x + t)

with generator A = ddx + -;. For-l < c < 0 and t > 0, T(t) is

not a bounded operator on X, but S(t) is, where

S(t)f = 1t T(s)fds.

This T is also an integrated semigroup.

We now give one of several equivalent definitions of an n

times integrated semigroup. The case of n = 0 corresponds to a

Co semigroup. A closed linear operator A is the generator of an


n-time.'J integrated .'Jemigroup on X ( for n = 0, 1,2, ... ) if for all

f E V(A), there is a unique wE C(lR+,X) such that J~tw(s)ds

belongs to V( A) and

1t tn w(t) = A w(s)ds + ,f

o n.

holds on IR+. If w E cn+l(lR+, X), then u = w(n) satisfies

u'(t) = Au(t), u(O) = f. (19)

For n = 0, one simply interprets this to mean Ut::C1(lR+,X) and

(19) holds.

It is a theorem that an operator A is the generator of an

n-times integrated semi group iff the initital value problem

du/dt = Au, u(O) = f

has a unique solution Ut::C 1(lR+,X) n C(lR+, V(A)) for all f E

V( A n+ 1 ) satisfying

n

Ilu(t)11 ~ Mewt 2: IIAi til i=o

where M ~ 1 and w E lR are constants, independent of f. Let A

generate an n-times integrated semi group on X. Then, according

to the theorem of Arendt, Neubrander, and Schlotterbeck, there

are Banach spaces Y, Z with continuous dense embeddings

so that the natural extension and restriction of A to Z and Y, Alz and Aly, both generate (Co) semigroups, on Z and Y respectively.

Thus is some sense, integrated semigroup theory reduces to (Co)

56 1. A. Goldstein

semi group theory. But in other ways, integrated semigroups are

indeed more general.

The theory of non (Co) semigroups is a rapidly developing

field; the theory is developing faster than the applications. In

teresting applications and examples have been obtained by W.

Arendt and by M. Hieber, and by O. Diekmann and his collabo

rators.

AFTERWORD.

We have made no effort to credit original sources or to give a

comprehensive bibilography. Most of these notes follow material

in my book [1]. For most of the rest of the material, see the papers

[2]-[7] and the references contained therein.

The author gratefully acknowledges the partial support of

the National Science Foundation.

The theory and practice of semigroups of linear operators

remains an active subject. It seems certain that in 1997 one will

be able to write a substantial survey paper on applications which

will be based on work done since 1992.

REFERENCES

1. J. A. Goldstein, Semigroups of Linear Operators and Appli

cations, Oxford U. Press, New York and Oxford, 1985.

2. J.A. Goldstein, Asymptotics for bounded semi groups on

Hilbert space, in Aspects of Positivity in Functional Anal

ysis (ed. by R. Nagel, U. Schlotterbeck, and M.P.H. Wolff),

Elsevier (North-Holland), Dordrecht (1986), 49-62.

3. J.A. Goldstein, Evolution equations with nonlinear bound-


ary conditions, in Nonlinear Semigroups, Partial Differen

tial Equations and Attractors (ed. by T. L. Gill and W. W.

Zachary), Lecture Notes in Math. No. 1248, Springer, Berlin

(1987), 78-84.

4. J. A. Goldstein (in collabration with W. Caspers, K. Engel,

M. Hieber, F. Rabiger, and G. R. Rieder), Lectures on Appli

cations of Semi groups of Linear Operators, Semesterbericht

Funktionalanalysis, Tiibingen (1990), 59-139.

5. J. A. Goldstein and J.T. Sandefur, Jr., Equipartition of en

ergy for higher order abstract hyperbolic equations, Comm.

P.D.E. 7 (1982), 1217-1251.

6. J. A. Goldstein and J.T. Sandefur, Jr., An abstract

d'Alembert formula, SIAM J. Math. Anal. 18 (1987), 842-

856.

7. J.A. Goldstein, R. de Laubenfels, and J. T. Sandefur Jr., Reg

ularized semigroups, iterated Cauchy problems, and equipar

tition of energy, Monatshefte Math. (1993).

Nonlinear Semigroups and Applications

GiseIe Ruiz Goldstein

1. Introduction Our aim is to study problems which are governed by the

abstract Cauchy problem

d:~t) = A(u(t))

u(O) = f.

t>O

(ACP)

Here X is a Banach space, u(t) takes values in X for each t > 0, and A is an operator (not necessarily linear) which maps its domain V( A) ~ X into X. We are interested in questions of existence and uniqueness of solutions (in some sense) of (ACP), as well as continuous dependence of solutions on the initial data. A problem which satisfies these three conditions is said to be wellposed. We shall also be concerned with qualitative features of the solution, such as regularity.

Problems which can be formulated as an abstract Cauchy problem arise in wide variety of applications, including physics, chemistry, fluid mechanics, mathematical biology, probability theory, quantum theory, and differential geometry. We mention a few important examples which fit into this framework.

Example 1: Let n be an open subset of lRn,n ~ lRn, with "nice" boundary an. The heat equation in n can be written as the initial value problem

av = ~v at

v(O,x) = f(x)

t>O (1.1 )

x E n.

The problem (1.1) may be studied with a wide variety of boundiary conditions; for definiteness we consider Dirichlet boundary

59

G. R. Goldstein and 1. A. Goldstein ( eds.), Semigroups of Linear and Nonlinear Operations and Applications, 59-98. © 1993 Kluwer Academic Publishers.

60 G. R. Goldstein

conditions

v(t,x)=O t ~ 0, x E an. (1.2)

Let X be a Banach space consisting of functions on n, and let A = ~. If we define the domain of A by V(A) = {w E X and w = o on an}, then this problem is of the form (ACP). One can prove that this problem is "good"; that is, it has a unique solution which depends continuously on the initial data in many spaces X.

Example 2: We consider the initial value problem for the wave equation on m.n .

Set

{ a2v A at2 = uV

v(O,x) = f(x) ~~(O,x) = g(x)

for t > 0, x E m. n

for x E m.n

for x E m.n .

( v(t,x)) (f(X)) (0 I) u(t,x)= Vt(t,x) ,F(x)= g(x) ,A= 6.0 '

(1.3)

and let u(t,·) E X, where X represents a space of pairs of functions on n. Then (1.3) formally reduces to the form (ACP). One can show that this is a "good" problem only in norms related to the energy norm, where

II (j:) II;nergy = 11\7!I11~2(1Rn) + Ilf211~2(lRn)" Here the first and second terms on the right side represent the potential energy and the kinetic energy respectively.

In Examples 1 and 2 the operator A is linear. We now give some examples which illustrate the importance of nonlinear operators A as well.

Example 3: Consider the initial value problem for the one dimensional Hamilton-Jacobi equation

{ av + H (aV) = 0 at ax

v(O, x) = f(x)

for t > 0 and x E m. (1.4)

for x Em..

Nonlinear Semigroups 61

Let X be a space of functions on IR. Define the operator A by A = - H 0 ddx • Clearly, A is nonlinear. This problem is "good" if we take X to be the space of bounded uniformly continuous functions on IR; it is not a nice problem in any LP norm for 1 ::; p < 00.

Example 4: Flow through a porous medium in a region n E IRn

can be modeled by the following initial-boundary value problem

{ ~~ = f1 (tp( v)) v(O,x) = f(x) v(t,x)=O

for t > O,x E n for x E n for x E an.

(1.5)

Let tp( v) = vm +1 = Ivlmv for m > 0, although one can allow for more general functions tp. As we shall see later, this problem can be put into the form (ACP); it is a "good" problem in either the space L1 (n) or H-1 (n).

Note that if m = 0, we have the linear heat equation. However, the solutions of (1.5) and (1.1), (1.2) have dramatically different features. In the case of the heat equation, with n = IRn , it is well known that the solution u may be written in the form

u(t,x)=(Gt*f)(x)

where Gt is the Gaussian heat kernel. Hence, if the initial data f(x) is positive on some set of positive measure, the solution u(t,x) > 0 for all x E JRn and all t > O. We say the equation has an infinite speed of propagation. In a later section we shall show that the porous medium equation has a finite speed of propagation.

Suppose that our abstract Cauchy problem is well-posed. Let {T(t) : t ~ O} be a family of operators on X, and suppose that T(t) maps the solution of (ACP) at time s to the solution at time t + s. The assumption that the operator A is independent of time is just the assumption that T(t) does not depend on s. The solution at time t + s can be represented by T( t + s)f where u(O) = f. On the other hand, if we start the equation at t = 0, let the system run for s units of time, then use the solution at time s as new initial data and run the sytem for t units of time longer, we see that by uniqueness we must have

62 G. R. Goldstein

T(t + s)f = T(t)T(s)f·

In addition we require T(O) = I and, t --+ T(t)f is differentiable for t 2: 0 and it T(t)f = AT(t)f, so that u(t) = T(t)f and T(t) is continuous on X; hence the solution depends continuously on the initial data. Finally, the initial data should be in V(A.). We assume V(A) = X. Note that T(t) is linear iff A is linear.

Consider the case where A (or equivalently T(t)) is linear. The family {T(t) : t 2: O} of bounded linear operators from X into X is a Co -semigroup (or strongly continuous semigroup ) if

{ T(t + s) = T(t)T(s) for all t, s E IR+

T(O) = I (1.6) T(-)f E cOO, 00); X) for all f E X.

The family {T(t) : t 2: O} ~ B(X) is a Co-semigroup of contractions if IIT(t)11 :s; 1. In fact suppose {T(t) : t 2: O} is a Co-semigroup. By the Uniform Boundedness Principle

L = sup IIT(t)1I < 00. 099

Let n = [t]; then T(t) = T(t-n)T(n) and IIT(t)1I :s; Ln+l :s; Lewt where w = lnL. Set

S(t) = ewtT(t);

then IIS(t)1I :s; L. If we define a new norm 111·111, on X by IIlflll = ~~~ IIS(t)fll, then

IIIT(t)flll = sup IIS(u)T(t)fll = sup lIe-WITT(u)T(t)fll IT>O IT>O

= sup lIeWITT( r )fll r>t

:s; sup lIe-w(r-t)T(r)fll = ewtlllflll· r>O

Hence, IIIS(t)1I1 :s; 1, and clearly 111·111 and II· II are equivalent norms. Roughly speaking, this calculation shows it is sufficient to consider only Co-contraction semigroups.

The following results are the basis for the theory of semigroups of linear operators.


Theorem 1: (Hille-Phillips) The abstract Cauchy problem (ACP) is "well-posed" iff it is governed by a Co-semigroup {T(t): t ~ O} iff A is the infinitesimal generator of a Co-semigroup T. Here Af = ~jlo T(t)[- f, and f E D(A) iff this limit exists. In this

case the unique solution of (ACP) is u(t) = T(t)f.

Formally this suggests that T(t) = etA where A = T'(O).

Theorem 2: (Hille- Yosida Generation Theorem) The operator A generates a contraction Co-semigroup T iff D(A) is dense in X and the following conditions hold:

11(1 - AA)-lll ~ 1 for all A > 0 (1.7)

R(I - AA) = X. (1.8)

The theory of semigroups has been extensively studied. Applications have been found in a wide range of areas. For more study of some aspects of the linear theory, the interested reader should refer to the preceding article by J. A. Goldstein as well as [18] [24] [28] [29]. One of the most amazing facts about the theory of semigroups of linear operators is that linearity is irrelevant. This outrageous statement is in some sense accurate. In the remainder of this paper we assume only that A is an operator on the Banach space X, where A is either linear or nonlinear.

Let us proceed formally. Consider the backward difference scheme for (ACP),

ue(t) - ue(t - E) _ A () -'--'--'--'-----'- - U e t .

E

Then (1.9) may be written as

{ue(t)= (I - EA)-lue(t - E)

ue(O)= f.

Letting E = ;, we see that

t ue(t) = (I - - A)-n f.

n

(1.9)

(1.10)

We are concerned with finding the limit of ue(t) as n ---t 00. In the applications usually A is some sort of differential operator

64 G. R. Goldstein

(and hence is unbounded). However, in that case the operators (I - EA)-l are integral operators which are smoothing.

On the other hand, if instead of (1.9) we use the forward difference scheme

(1.11)

then the analogue of (1.10) is

(1.12)

Bu the operators (I + ~ A) -n are differential operators; hence ue(t) doesn't converge in general even in the linear case.

Our first goal is to find a nonlinear analogue of the HilleYosida theorem. We define the Lipschitz seminorm of an operator S on X by

IISliLip = inf{C E IR+ : IISf-Sgil ~ Cllf-gll for allf,g EX}.

If II (I - '\A)-l IILip ~ 1, we say A is dissipative. If A is dissipative and R(I - '\A) = X for some ( and hence all) ,\ > 0, then A is m-dissipative. We define the semi group T = {T(t) : t E IR+} generated by A via the formula

T(t)f = lim (I - ~A)-n f n-+oo n

(1.13)

for all f E V(A). T is a contraction semigroup if IIT(t)IILip ~ 1 for all t :2: O.

The following theorems form the nonlinear analogues of the Hille-Yosida and Hille-Phillips theorems.

Theorem 3:(Crandall-Liggett) If A is m-dissipative on X) then A generates a strongly continuous contraction semigroup on V(A).

Theorem 4: (Benilan-Kobayashi) If A is m-dissipative) then (A CP) is well-posed.

Note that in the nonlinear case the implications are only in one direction; it is, however, the most important direction for the applications.

Let us make a few comments about these theorems. The semigroup of Theorem 3 is given by

T(t)J = lim (I - ':A)-n J n-oo n

for J E X,

and the solution of (ACP) is given by u(t) = T(t)f. In general u may be nowhere differentiable, but if X has the Radon-Nikodym property, then u'(t) E Au holds a.e. for J E V(A). (Recall that all reflexive spaces have the Radon-Nikodym property). For a general Banach space X the notion of solution is in a very general sense, which we explain in the next section. Hence, the notion of well-posedness in Theorem 4 must be general enough to allow for this notion of solution.

Finally we note that the converse of Theorem 3 is false in general, but holds in the Hilbert space case.

This paper is based on the short course which I presented in Cura<;ao. I warmly and sincerely thank Jerry Goldstein for his lecture notes, which are soon to be a book [19] on this topic. Sections 1-5 of this paper follow closely Jerry's notes.

§2. The Notion of Solution-An Introduction Let H be a Hilbert space, and let A be a linear operator

on H. Then the condition (1.7) in the Hille-Yosida theorem is equivalent to

Re(AJ, J) :::; 0 for all f E V(A). (2.1)

The corresponding condition in the nonlinear analogue of the Hille-Yosida theorem is

(2.2)

( Here A may be nonlinear). If A is linear, clearly (2.2) reduces to (1.7). In general (2.2) is equivalent to

Re(AJ - Ag,J - g) ::; 0 for all J, 9 E V(A). (2.3) (I}A)/ •

'--------~ ~-~---~------>

Figure 1. x Figure 2. x

66 G. R. Goldstein

Let us look at the example A : IR -+ IR, where A is a singlevalued function of one (real) variable. Clearly (2.3) holds iff A is a nonincreasing function. Since A is not necessarily continuous, the range of (I - '\A) is not necessarily all of IR. (See Figures (1) and (2)).

In order to make (I - ,\A)-l everywhere defined, we must "fillin" the gaps, that is to say we must allow the function A to be multivalued. If A has a discontinuity at Xo, we redefine A so that its value at Xo is the interval [A(xo+),A(xo-)] where A( Xo ±) = lim A( x) Allowing A to be multi valued allows the

x ...... x± o range condition, R(I -'\A) = X, in the Crandall-Liggett Theorem to be satisfied.

We begin our discussion of the notion of solution for the nonlinear problem (ACP) by considering some examples. \Ve begin with a nonlinear conservation law in one dimension

{ ~~+:X(~(U))=O u(x,O) = f(x)

x E IR, t > 0

where ~ : IR -+ IR is a smooth function.

(2.4)

Suppose u is a classical solution of (2.4). Then on the characteristic curves x( t), defined by

dx ( ) dt = ~' u (.T(t), t) ,

we have, by the equation,

! (u(x(t), t)) = Ux ~: + Ut = ux~/(U) + Ut = O.

Thus, u is constant along characteristics, and it follows that ~~ is constant. That is, characteristics of (2.4) are straight lines in the (x,t) plane. (See Figure 3.)

In order to satisfy the initial condition, we must have u(xo, to) = f(x*). In general characteristics can and do intersect, so we can't have classical solutions to (2.4). (See Figure 4.)

Nonlinear Semigroups

t)...

!,td 1--........ ---->

(X'·,O)

Figure 3

x

t)...

(x • t ) o 0

1-----<1_-->-(x·,O) (X, 0) x

Figure 4

67

For example in one space dimension if we consider the Hopf equation Ut + UU x = 0 with u(xo, to) = J(x), then the characteristic equation is

dx - = u(x,t). dt

Hence ep'( u) = u, so ep( tL) = u22 ; in particular ep is convex. Let

x* and x be as in Figure 4. Clearly the characteristics through (x*,O) and (X,O) intersect whenever J(x) < J(x*). If we want to allow initial data with compact support, then this example shows we cannot expect classical solutions. Hence, if we want global solutions we must weaken our notion of solution to allow for discontinuous solutions.

We can define solutions to (2.4) by multiplying the equation by 9 E COO(JR) and integrating by parts. A measurable function u E Lool (JR) is a weak solution of (2.4) if oc I: 100 [-gtU - gxep(u)] dtdx + I: J(x)g(x,O)dx = O. (2.5)

This condition is sufficiently weak to allow for the existence of global solutions. However, solutions defined by (2.5) are not unique. For example if we again consider the Hopf equation we can easily verify that

x<! 2

x>! - 2

u,(t) = {1 x~O O<x<t

x? t

68 G. R. Goldstein

are both weak solutions, with the same initial data; in fact there are infinitely many weak solutions with this data. Somehow we must find a way to single out the physically correct solution.

Oleinik's idea was first to regularize the problem (2.4), that is, find ue ( for 6 small) which satisfies

{ Ui + cp (ue)x = 6U~x (2.6)

ue(O,x) =f(x).

Set u(t, x) = lim ue(t, x). Then this solution exists; this method e->O+

is known as the method of vanishing viscocity. We shall require a further condition, known as an "entropy condition", to guarantee uniqueness of a generalized solution.

Let 9 E Cgo(IR x [0,(0)) and cP E C2(IR). It is easy to see that

cp( u) = cp' ( U )ur for r = x, t,

and for k E IR,

cp'(U)uxcp(u) = [l U cpl(S)CP/(S)dsL·

Multiplying the equation (2.6) by gCP( u) and integrating by parts yields

I: = f L: [~(U')9t+ g. (( 1"'(S)~'(S)dS) 1 dxdt

= 6 100 I: [U~gxcp/(U) + (u~?gcplI(ue)] dxdt

= -6 [00 100 cp( ue)gxx + 6 [00 100 (u~)2 gcpll( ue). 10 -00 10 -00

The last term on the right side involves (u~)2 which is difficult to estimate. However, if we assume 9 is nonnegative and cp is convex, then this term is nonnegative. Thus,

I '2 -6 [00 100 cp( ue)gxx, 10 -00

and letting 6 ! 0, we see that 1'2 0.

Nonlinear Semi groups 69

Actually, it is sufficient to consider the function ~(s) = Is-kl. This method, developed by Kruzkov, gives a notion of solution to (2.4) which is defined by a family of inequalities. This solution (u = lim ue') exists, is unique and may be discontinuous. For our example of a single conservation law, a solution defined in this way allows for downward jumps only, provided <p is convex.

Notice that if v is a classical solution of the Hamilton-Jacobi equation (1.4) and if we set u = ~;, the Hamilton-Jacobi equation reduces to (2.4) with H = <po Solutions of the Hamilton-Jacobi equation may therefore be expected to be continuous, but their spatial derivatives may have discontinuities.

The physically correct space in which to consider the Hamilton- Jacobi equation is a space involving the supremum norm, while for the conservation law the correct space is Ll(IR). Hence we must allow the theory of nonlinear semigroups to be general enough to include the notion of solution indicated above, and we must allow for nonreflexive Banach spaces.

3. A sketch of the proof of the Crandall-Liggett Theorem Let X be a Banach space and D a closed subset of X. We

write T E Sw(D) if T is a strongly continuous semi group of type wand, that is IIT(t)IILip :S ewt. An operator G is the infinitesial generator of T if

Gf = lim T(t)f - f tlO t

(3.1)

where D( G) consists of all functions fED for which this limit exists. In the nonlinear theory the operator G may be empty. For an example let Vet) be a unitary group on a Hilbert space H, and let A be the generator of the unitary group on H. Suppose that A is unbounded, A-I exists as a bounded operator and choose f ~ D(A). Set D = {U(t)f : t E IR}. Clearly, D is a closed subset of H but not a linear space and UID E So(D), but D(A) n D = </>. Thus G = </>.

H G is a subset of X, we define

IGI = inf Ilfll· fEG

(3.2)

Let A ~ X x X is a multi valued function from X to X, that is Af = {g EX: (j,g) E A}. The domain of A, D(A), is

70 G. R. Goldstein

{f : Af i= 0} and the range R(A) = U{Af : f E V(A)}. If B ~ X x X is a multivalued function and .x E 1R or <C, then A + B = {(f,g + h): (f,g) E A and (f,h) E B},.xA = {(f,.xg): (f,g) E A} and A-I = {(g,j): (f,g) E A}. We identify functions with their graphs.

An operator A on X is dissipative if II (I - .xA)-IIILip :::; 1; A is m-dissipative if A is dissipative and R(J - .xA) = X holds for some.x > O. We say A is essentially m-dissipative if A is dissipative and R(I - .xA) = X, for some .x > O.

Let X* be the dual space of X. The normalized duality map J : X ~ 2x * is defined by

J(x) = {x* E X* : (x,x*) = Ilxll, IIx*1I = I}. (3.3)

Hence, 'P E J(f) iff (j, 'P) = IIfll II'PII = Ilfll. If X = H, a Hilbert space, then J = niTJ (except at the origin) where I: H ~ H is the identity. If X = C[O, 1], the real space, then for f E X, there is a ~ E [0,1] such that II'PII = ±'P(O· Then 'P E J(f) if (j,'P) = IIfil = ±f(O, hence 'P = ±8e· From the linear theory we have the following lemma.

Lemma 3.1: Let f, g, E X. Then IIfll :::; IIf - ag II for all a > a iJJthere exists 'P E J(f) satisfying Re(g,'P):::; O.

If we apply this lemma to the nonlinear case we obtain an alternate definition of dissipativity.

Lemma 3.2: Lei A be a (possibly) multivalued function on X x X into X. Then A is dissipative iff for all UI,U2 E D(A) and with Vi E AUi, there exists 'P E J( UI - U2) such that

Re(vI - V2, 'P) :::; o.

This lemma follows easily from Lemma 3.1 by noting that A is dissipative iff

for all hI, h2 E R(I - .xA) and for all .x > O. Setting Ui = (J -.xA)-1 hi, and Vi E AUj, we see that

IIuI - u211 :::; lIuI - .xVI - U2 + .xv211

= II (UI - U2) - .x (VI - V2) II,

so Lemma 3.2 follows from Lemma 3.1. One can also show that if A is m-dissipative, then R( I -

AA) = X for all A > o. Before beginning our outline of the proof of the Crandall

Liggett Theorem we make a few remarks. First, we shall allow for multivalued operators A. In addition the Crandall-Liggett Theorem remains valid if we only assume that A-wI is dissipative for some w E ffi and that there exists AO > 0 such that R( I -AA) J V( A) for all 0 < A ~ AO. In this case A generates T E S",(V(A)) via the formula (1.13).

Outline of the Proof of The Crandall-Liggett Theorem.

We outline the proof in the case w = o. We assume that 11(1 - AA)-lIILip ~ 1, and we must show that

J~nl = (I - ;A) -n 1-+ T(t)1

as n -+ 00. The main step is thus to show that {J~nl} is a

Cauchy sequence for I E V( A) So for n ~ m and 0 < fJ ~ A, we define h. = (I - AA)-l and am,n = IIJ;I - Jf III.

Recall the resolvent identity for linear operators A: for S" > 0,

Its analogue for nonlinear operators is

for A, fJ > o. Then

am,n = IIJ; 1- Jf III = II J; 1- J" (aJ~-l 1+ (3Jf I) "

where a = X and (3 = A~". Hence

since IIJ"IILip ~ 1. Thus, we have

am,n ~ aam-l,n-l + j3am,n-l. (3.4)

72

n

Figure 5.

G. R. Goldstein (m ,n+l)

(m,n)

(m,n- t)

m

Identify am,n with the weight associated with the point (m, n). As the above figure indicates, if we start with am,n and repeatedly apply (3.4), we will end up with terms involving ak,O

or aO,j together with certain coefficients. More specifically m-I n-m

a < "C . a'O + "D . a . m,n - D m,n,)), D m,n,) 0,)

i=O i=O

where Cm,n,i and Dm,n,i are combinatorial-type coefficients. vVe estimate the remaining coefficients by

i-I

aO,i = Illtl - III ::; L IIJ!+1 1- J;/II k=O

i-I

::; j L IIJI'I - fll k=O

::; jpIIAIII·

Here, for simplicity, we have assumed that A is single-valued. The last step follows from

IIJI'I - III = IIJI'(u - pAu) - JI'U - pAJ)1I ::; 1I1L - pA.lL - f + pAfll ::; pllAfll

if (I - p.4)-1 1= 1L. vVe can bound the weights aj,O similarly. 0

Again we note that the range condition maybe weakened to R(I - AA) = X or R(I - AA) 2 V(A), and the Crandall-Liggett theorem still gives a semigroup via the formula

t -T(t) = lim (I - _A)-n. n--+oo n

We also note that the Crandall-Liggett theorem will give local solutions where they exist. For example, consider the equation u' = u2 with the initial condition u(O) = ~ with c > O. Then u( t) = t-=-lc' so there is a smooth solution which exists up to time c where c can be arbitrarily small. The Crandall-Liggett theorem gives the formula for this solution, where it exists.

In the linear theory, we can assume

IIT(t)11 :::; Mewt

for some M > 1 and w > O. There is no analogue fo this in the nonlinear theory. Recall that the Crandall-Liggett proof required "stripping off" powers of J: one at a time. If IIJ:IILip ::; M with M > 1, then these powers of Jvl build up, and the proof breaks down. In fact, the Lipschitz condition for general wand f E V(A) IS

We write A E Gw for wEIR if A - wI is closed and dissipative and there is a Ao > 0 such that R(I - AA) :J V(A) for 0 < A ::; Ao.

We write A E Gwif A - wI is dissipative and R(I - AA) = X for small A > O. Then A generates a semi group of type W by the Crandall-Liggett theorem, and A is closed. If {T(t)} is the semigroup generated by A E G-::;, we write T E Sw((V(A)). Clearly, if B is a globally Lipschitzian operator, then B E G~BIILip' and

A + B E G~+IIBIIL' . zp

We shall return to perturbation theorems in a later section.

4. The Notion of Solution In this section we discuss the sense in which the semigroup

given by the Crandall-Liggett theorem determines a solution of the abstract Cauchy problem

u'(t) E A(u(t)) u(O) = f

O:::;t<oo f E V(A).

(ACP)

A function u is a strong solution of (ACP) if u is locally absolutely continuous from IR+ in X, u(t) E V(A) for almost every t, u'(t) E Au(t) for almost every t, and u(O) = f.

74 G. R. Goldstein

The following theorem shows that if (ACP) has a strong solution, then it is unique and the semigroup of the Crandall-Liggett theorem gives it.

Theorem 4.1: Suppose A E G::;. Consider (A CP) for f E V( A). Then for a function u : ffi.+ ---+ X the following are equivalent:

(i) u is a strong solution of (A CP).

(ii) u is strongly differentiable a.e., and u(t) = T(t)f for all t ~ O.

Corollary: Suppose A E G::; and suppose X is reflexive, then (A CP) has a strong solution which is unique.

The proof follows from the Fundamental Theorem of Calculus and the preceding theorem. Note that if we assume only A E Gw ,

then by the Crandall-Liggett theorem A generates a semigroup via the exponential formula, so for f E V(A), T(t)f is well-defined. Theorem 4.1 remains valid for the solution u(t) of (ACP) for 0 ::; t < T. We also note that the Corollary remains valid whenever X has the Radon-Nikodym property. However, many spaces which are important in applications do not have the Radon-Nikodym property, such as Ll (ill.) or C[O, T] equipped with the supremum norm. In nonreflexive spaces it is possible for (ACP) to have a "solution" which is not weakly differentiable at a single point.

We want a notion of solution in a general Banach space X so that we have existence, uniqueness and continuous dependence on initial data for (ACP). There are two such notions of solution; these notions are equivalent when A E Gw • The first notion, based on differences schemes, is due to Kobayashi [25] with substantial extensions and improvements due to Evans [14]. The second notion of solution (which is the first historically) is due to Benilan [5]; it is based on a family of inequalities.

We consider the abstract inhomogeneous Cauchy problem

{Ul(t) E Au(t)+g(t) 0::; t ::; T

u(O) = f. (AICP)

Here T > 0, possibly T = +00. Let g E Ll([O, T]; X) and f E V(A). For each n E IN, we consider a partition 0 = to' < ti < ... < tN(n) = T, and let xi: E V(A), gi: E X for n E IN and


k E {O, ... N(n)}. A backward difference scheme for (AICP) is a series of triples {(tl:,xk,gk): n E IN and k E {O, ... ,N(n)}} which satisfies the following properties.

( 4.1)

for n E IN and k = 1, ... , N ( n );

lim max (tk - tk- 1) = 0; (4.2) n-+CX) k

lim Ilx~-fll=O; (4.3) n-+CX)

iT Ilgn(t) - g(t)lldt -+ 0 as n -+ 00. (4.4)

where

() { gkn for t E (tkn -1' tnk] gn t = go for t = o.

N ow we define

If un(t) -+ u(t) uniformly as n -+ 00, and u(t) is continuous, then we say u(t) is the limit solution of (AICP) on [0, T].

Notice that (4.1) may be rewritten as

xl: = (I - b;:A)-l (Xk-1 + bl:gl:) (4.5)

where bl: = tk-tk- 1 is the length of the kth time step. If A E G~, then the range condition R(I - AA) insures that the mapping in (4.5) is well-defined.

For limit solutions existence is easy to prove, but uniqueness is difficult. In addition it is not easy to say in the sense which limit solutions are differentiable. Often we require other techniques to prove additional regularity.

To motivate the second notion of solution, we first consider the case of X a Hilbert space. Let u be a strong solution of

76 G. R. Goldstein

(AICP), and suppose that A - wI is dissipative and single-valued. Then if x E D(A)

~~llu(s)-xW = ~:s{u(s)-x,u(s)-x) = Re{u'(s),u(s) - x) = Re{Au(s) + g(s), u(s) - x) = Re{Au(s) - Ax, u(s) - x) (4.6)

+ Re(Ax + g(s), u(s) - x) S; wllu(s) - xW

+ Re(Ax + g(s), u(s) - x).

That the second equality in (4.6) holds is an easy lemma. Integrating (4.6) from s = r 2: 0 to s E [r, TJ we get

1 1 jt 211 u ( t) - x W S; 211 u ( r) - x W + W r II u ( s) - x W ds

+ jt Re(g(s) + Ax,u(s) _ x)ds. (4.7)

We say u is an integral solution of (AICP) if u( t) satisfies (4.7) for o S; r S; t S; T. In order to extend this notion to Banach spaces note that the equality ~ tsllu(s) - xW = Re(u'(s),u(s) - x) is true in a general Banach space in the form ~ Lllu(s) - xl1 2 = Re(u'(s), </J} for any </J E J(u(s) - x).

Clearly integral solutions are not unique as the next example shows.

Consider (AICP) with the operator A = ~ and 9 == 0, on the space X = L2(S2) and D(A) = C~(S2). Let S2 be a smooth bounded domain IRn and suppose AD = ~ with Dirichlet boundary conditions and A. N = ~ with Neumann boundary conditions. The solutions, exp (tAD)f and exp (tAN )f, of these problems are different, but both are integral solutions of (ACP). Hence, we must require more if our new notion of solution is to be unique.

For x, y E X, we define

(x,y)s = sup{Re(x, </J): </J E J(y)}. (4.8)

Suppose A - wI is dissipative. Then a continuous function u : [0, TJ -+ X for X a Banach space is an integral solution of (AICP)


on [0, T] if u(o) = f and

1 1 jt 21Iu(t) - xW ~ 2"u(r) - xW + W r IIU(T) - XWdT.

+ 1t(9(T)+y,U(T)-X)S

( 4.9)

holds for all ° ~ r ~ t ~ T all x E V( A) and for all y E Ax. A continuous function U : prO, T] -t X is called a mild solution of (AICP) on [0, T] if u(O) = f and for all integral solutions of

we have

{VI = Av + h(t)

v(O) = f

1 1 2"u(t) - v(t)W ~ 2"u(r) - v(r)1I2

on [0, T]

+ 1t (9(T) - h(T),U(T) - v(T))sdT

for ° ~ r ~ t ~ T.

( 4.10)

(4.11 )

One can easily show that if u and v are strong solutions, then (4.11) holds. Also, if y E Ax and if we set h(t) == -y and v(t) == x, it is clear that a mild solution is an integral solution. Mild solutions satisfy a Gronwall-type inequality; hence, uniqueness, for mild solutions is easy to prove. However, existence is difficult.

Theorem -4.2: Consider the abstract inhomogeneous Cauchy problem (AICP). Supp~A - wI is m-dis8ipative 9 E Ll([O, T]; V(A)) and f E V(A). Then (AICP) has a unique limit solution on [0, T] which coincide8 with its unique mild solution on [0, T]. When g(t) == 0, the solution i8 given by u(t) = T(t)f, where T(t) is the semigroup generated by A-wI via the CrandallLiggett theorem. The solution u( t) exi8t8 for all t E IR+ provided 9 E Ll([O,T];V(A)) for allT > 0.

We close this section with some brief observations concerning the question of differentiability of the semigroup.

Suppose A generates a semigroup of type w. The Yosida approximation A,\ of A is defined by

(4.12)

78 G. R. Goldstein

for A > 0 and f E V( A). We define the functional

N(f) = lim IIAAfl1 ALO

(4.13)

which exists in [0,00]. It is not difficult to see that N(f) < 00 may be viewed as the condition that the semigroup T E Sw(V(A)) applied to f satisfies a Lipschitz condition with respect to t. More specifically, we can show that

N(f) = lim c11IT(t)f - fll; t!O

( 4.14)

that is, one side of (4.14) is finite iff the other is and equality holds.

We define the Favard class (or the generalized domain) of A to be

V(A) = {/ E V(A) : N(f) < oo}.

We write A E G~ax if A E Gw and A-wI has no proper dissipative extension in X. Clearly,

Gm c Gmax c G w w w,

and if A E Gw , then A may be extended to Ao E G~ax by Zorn's lemma. It is also clear that

V(A) ~ V(A) ~ V(A).

The following theorem summarizes the differentiability properties of the semigroup.

Theorem 4.3: Let A E G~ax, and let T E Sw(V(A)) which A generates.

(i) For all t E IR+,

T(t)(V(A)) ~ V(A).

(ii) Iftn ~ 0 as n ~ 00 and

g = w - lim n--oo

where f E V(A), then g E Af and

N(J) = Ilgll = lim t;;lll T(tn)J - fll· n--oo

(iii) If X is reflexive, then V(A) = V(A) {f E V(A) as t -t 0, IIT(t)f - fll = O(t)} and for all f E V(A),

N(J) = limC11IT(t)f - fll· f!O

In a nonreflexive Banach space, (i) can fail to hold if we

replace V( A) by V( A ).

5. Approximation and Perturbation Theory

The following hypothesis will be assumed throughout this section

Assume

An - wI is dissipative and R(I - AAn) ;2 V(An) for each (HYP)

n E INo and for 0 < A < AO, where AO is independent of n.

In particular (HYP) implies that An E Gw for each n E INo. Let Tn E Sw(V(An)) be the semigroup determined by An via the exponential formula.

We are interested in theorems which give continuous dependence of the solution un(t) of

{ U~(t) E Anun(t)

un(O) = fn

on both the initial data fn and t11f' operators An.

(ACP)n

Theorem 5.1: Let (HYP) hold. Suppo,qe that for each go E V(Ao), there is a gn E V(An) such that gn -t go as n -t 00.

ASS1lme that if f n E V( An) and f n -t fo, then

as n -t 00 (5.1)

80 G. R. Goldstein

for 0 < ). < ).1 (where ).1 is independent of n). Then

lim Tn(t)fn = To(t)fo (5.2) n----oo

whenever fn E V(An) and fn ~ fo. Moreover the convergence is uniform for t in compact subsets of lR,+.

Theorem 5.2: Let (HYP) hold, and suppose that V(Ao) C V(An) for all n E INo and for all fo E V(Ao),

as n ~ 00 (5.3)

for 0 < ). <).1 (where ).1 is independent of n). Then

lim Tn(t)fn = To(t)fo n----oo

whenever fn E V(An) and fn ~ fo, the convergence being uniform for t in compact subsets of lR,+.

In the linear case (that is An and Tn(t) are linear and -=-:--:---.,.-V(An) = X), the statements

Tn(t)fo ~ To(t)fo asn~oo (5.4)

and

(1 - )'An)-1 fo ~ (I - )'Ao)-l fo asn~oo (5.5)

are equivalent, while (5.5) implies

(I - )'An )-1 fn ~ (I - )'AO)-l fo as n ~ 00. (5.6)

It follows by Theorem 5.1 that (5.6) implies (5.4). Thus, (5.4) is equivalent to (5.6), and the hypotheses in Theorem 5.1 are "best possible" in this sense. That (5.4) implies (5.6) can be seen by looking at

and using the dominated convergence thoerem.

In fact we will now show that Theorem 5.2 is a special case of Theorem 5.1. We shall require the following fact: If A - wI is dissipative, then

(5.7)

for 0 < A < w -1 .

Now assume the hypotheses of Theorem 5.2, and let fn E -=-O-~

V(An),fn ~ fo and 0 < A < AI. Then

11(1 - AAn)-1 fn - (1 - AAo)-1 foil ::; 11(1 - AAn)-1 fn - (1 - AAn)-1 foil

+11(1 - AAn)-1 fo - (I - AAo)-1 foil. (5.8)

But since An - wI is dissipative,

and by hypothesis (5.3),

11(1 - AAn)-l fo - (1 - AAo)-1 foil ~ 0 as n ~ 00.

Hence, both terms on the right side of (5.8) go to 0 as n ~ 00,

that is (5.1), and hence the hypothesis of Theorem 5.1, holds.

Next we briefly describe some of the main perturbation theorems for nonlinear semigroups. Our first perturbation theorem is a nonlinear analogue of a standard perturbation theorem in the linear theory.

Theorem 5.3: Let A E G::1 , and let B - w2I be dissipative and single valued with V(A) C V(B). Suppose there are constants o < a < 1 and b ~ 0 such that

IIBf - Bgil ::; al·4f - Agi + bllf - gil (5.9)

holds for all f, g E V(A). Then

82 G. R. Goldstein

Corollary: If A is m-dissipative and B is globally Lipschitzian, then A + B E G~BII ..

Lzp

Sketch of Proof of Theorem 5.3: Let 0 < a < ~._We assume,

for siElPlicity, that A and B are single-valued. Let A = A - WII and B = B - w2I. Then

II.8f - .8gl1 ::; II Bf - Bgil + IW2111f - gil ::; aliAf - Ag - wI! + wlgl1

+ (b + IWll + IW21)lIf - gil

= allAj - .4gl1 + b111f - gil.

Next, .4+.8 is dissipative. It remains to show R(I -A(.4+.8)) = X for some A > O.

Choose A > 0 so large that (2a + b1 A-I) < 1 which is possible since a < ~. Then

11.8(,\1 - .4)-1 f - .8(A - .4)-lgl1 (5.10)

Set

Then by (5.10)

::; all_4(AI - .4)-1 f - .4(AI - .4)-1 g l1

+ b111(AI - .4)-1 f - (AI - .4)-1 gil

::; aIlA(AI - .4)-1 f - A(AI - .4)-lg - (J - g)1I

+ b111(,\I - .4)-1 f - (,\1 - .4)-1 gil S; (2a + bI A -1 )111 - gil.

IIS9ft - S912 II ::; (2a + bl A -1 ) 11ft - f211.

Applying the Picard-Banach fixed point theorem we obtain a unique fa E X such that

fa = 9 +.8(I - A.4)-1 fa. (5.11 )

Setting u = (I - A.4)-1 fa, we see that (5.11) is equivalent to

- -Au - Au = g + Buo,

that is, g = ()"I - (A + B))uo. Hence g E R()"I - (A + B)), and since g was arbitrary, we are done.

For the case ! < a < 1, consider A + aB which is mdissipative for 0 ~ a ~ (2a)-1 by the preceding argument. The result follows by regarding j3B as a perturbation of A + aB and iterating this procedure.

Theorem 5.4: Let A - wlI be a densely defined linear mdissipative operator on X. Let B : X ---t X be everywhere defined and continuous, and suppose B-W2I is dissipative. Then A+B E

G:I +"'2 . Moreover, if T( t) is the semigroup determined by A + B via the Crandall-Liggett theorem, then the strong infinitesimal generator of T(t) is precisely A + B.

The proof of this theorem is technically difficult, but the idea is simple. Let S = {S(t) : t E IR+} be the linear semigroup generated by A. Then the main idea is to show that the semigroup T given by the variation of parameters formula

T(t)f = S(t)f + lot S(t - s)BT(s)fds

is well defined, holds globally in t and that its generator is A + B.

6. Applications of the Theory.

There are many important examples which can be solved via semi group theory. We shall consider two of them here.

Spatially Degenerate Parabolic Problems

We consider equations of the forms

Ut = Au. (6.1)

Such equations are often governed by contraction semigroups on LP for all p, 1 ~ P ~ 00 if the operator A is a uniformly elliptic second order operator. Here we consider such equations where the operator A is allowed to degenerate at the spatial boundary. We shall give the arguments in one space variable for simplicity. We shall then state, but not prove, some known results in higher dimensions.

84 G. R. Goldstein

We consider the following initial - boundary value problem:

Ut = <p(x, Ux)U xx + 'ljJ(x, u, Ux) u(x,O) = f(x)

u(O, t) = (Wx(O, t) u(l, t) = -j3ux(O, t).

(6.2)

Here x E [0,1], t E IR+, and a.,j3 ~ o. We make the following assumptions on o for x E (O,l),e E 1R (6.2)

and 0 ~ <po(x) ~ <p(x,e) where <POl E LI[O, 1].

'ljJ E C([O, 1] x IR2), and there exists J{ > 0 such that

There exist functions 9J1,.i! : IR -+ 1R which are contintLOUS and nondecreasing, on [O,oo} and .i!(r) ~ L(l + Ir!) for some L > 0 and

1'ljJ(x, e, 1])1 ~ 9J1(lel)(l + <po(x)).i!(l1]I). (6.4)

We take X = C[O, 1] equipped with the supremum norm. We consider the abstract Ca.uchy problem

{ut = Au(t)

u(O) = f(x)

where the operator A is defined by

(Au )(x) = <p(x, u')u" + 7,b(x, u, u')

with domain

V(A) = {u E C 2(0, 1) n CI[O, 1] : Au E C[O, 1] and u(O) = a.u'(O), u(l) = -f3u'(l)}.

(6.5)

The following claim follows from the second derivative test.

Claim: If u E C2( 0, 1) n C1 [0,1] satisfies the boundary conditions in (6.2), then lIull i= ±u(O for e = 0,1. Hence, if lIull = ±u(O, then 0 < e < 1 and

u'(O = 0, =fu"(O ~ O.

Our strategy is to show that A - wI is m-dissipative for some w > 0, so that by the Crandall-Liggett theorem and the results of Benilan-Kobayashi we obtain a unique mild solution of (6.5). Hence, in a certain sense, we have solved (6.1). Additional regularity results from the mild solution follow from results of [13]. We shall state those later.

Lemma 1: A - wI is dissipative for w ~ I< (where I< is as in (6.9}).

Proof: We wish to show t.hat

(Au - Av, 1» ::; wllu - vii for 1> E J( u-v). Since we are using the supremum norm, 1> E J( w) iff <P = ±8e where 111011 = ±w(O· Suppose that lIu - vii = (u -v)(e). To avoid trivialities, we assume that u i= v. By the claim we have 0 < e < 1, u'(O = v'(O and u"(O ::; v"(O

(Au - Av, 1» = ('P(e, u'(e))u" (0 - 'P(e, v'(e)))v" (e) + 1/;(e, u(e), u'(O) -1/;(e, v(e), v'(O)

= 'P(c, u'(t))[u"(O - v"(e)]

if w ~ I<.

+ lN~, u(O, u'(O) -1/;(e, v(e), u' (e)) ::; 1(( 1l - v)(O ::; wllu - vii

Theorem 6.1: A - wI is m-dissipative for w ;:::: I<.

Proof: First we prove the result in the case A is dissipative, that is, w = 0. This holds, for example, if 'IjJ( x, e, 7J) is either independent of e or is non decreasing in e.

By Lemma 1, it suffices to show R(I - >.A) = C[O,I] for >. sufficiently large, that is given h E C[O, 1] we must find a function u E V(A) with u = (I - >'.4)-1 h. Hence, u must satisfy

{u - >''P(-, u')u" - >'1/;(., u, u') = h

(6.6) u(O) = au'(O),u(l) = -,Bu'(I).

86 G. R. Goldstein

Let G be the Green's function for

{ ull = h (6.7) u(O) = au'(O),u(l) = -fiu'(1).

Define the operator S: C[O, 1] ~ C[O, 1] by

Su(x) = t G(x, y) u(y) - hey; (A¢~r)~(Y)' u'(y)) dy. (6.8) io rp y, u y

Then solving (6.6) is equivalent to finding a fixed point u of S. To accomplish this we use the next theorem.

Theorem 6.2: (Schauder Fi~ed Point Theorem) Let X be a Banach space. If T : X ~ X is continuous, compact and leaves some closed convex set invariant, then T has a fixed point in X. That is, there exists x E X such that Tx = x.

Clearly, S : C[O, 1 J ~ C1 [0,1] and S is continuous. Let Su = w. Consider the case of sublinear growth of ¢(x,~, 1]) in 1], that is, suppose

I¢(x,~, 77)1 ~ 9J1(1~I)(1 + rpo(x )),c(I1]i) (6.9)

where ,c(I)) ~ a as 11]1 ~ 00. Then I)

Ilwll ~ IIGIIA-Illrp;llh(llull + Ilhl!) (6.10)

+ IIGII9J1(llull)(l + Ilrpollld,c(llu'll)

follows from (6.3), (6.8) and (6.9). Also, since

" u-h-AljJ(·,u,u') w = Arp(.,U') ,

we see that

IIw"III ~ A-I IIrp;1 IIl(lIuli + IIhll) (6.11)

+ 9J1(lIull)(lIrp;1111 + l),c(lIu'II)·

Claim: If IE C1(0, 1) n C2 [0, 1] with fll E L1[0, 1] then

Ill' II ~ 411111 + 11111 111. (6.12)

The proof of this claim is easy; it can be found in [13]. Moreover it follows that {(SuY : u E B} is a equicontinuous

collection if B is bounded in e 1 [0,1]; hence, by the Arzela-Ascoli theorem, S : C1 [0,1] -t e1 [0,1] is compact.

Choose No so large that No ~ 511hll and

Choose>. > 16(1+IIGII)II'Po1 Ih and Mo = IIhll+No. We consider the closed, convex bounded set

B = {u E e1 [0, 1] : lIuli S; Mo, lIu'li S; No}.

Then by (6.10) and (6.11) we have

1 . 1 1 IIwll S; 16 (Afo + IIhll) + 16 No S; SMo,

IIw"lll S; 116 (j\I[o + IIhll) + 116No S; ~Mo.

Using (6.12) we see that

IIw' ll S; 411wll + IIw"1l1 S; ~Mo + ~1YJo < No,

whence, by the Schauder fixed point theorem S(B) C B. In order to handle the case w > 0, we must proceed in two

steps. First, for each v E e[O, 1], we consider the operator

Clearly, A v is dissipative; hence, J~ = (I - >.Av )-1 is well-defined, exists and is single-valued by the preceding argument. Next, for each h E C[O, 1] we consider the operator S : C[O, 1] -t e[O,l] defined by

ShV = JXh.

Finding a fixed point u of Sh is equivalent to showing that u ->.Au = h. The details of finding a fixed point of Sh for a < >. S; >'0, where a < >'0 < w- 1 can be found in [13].

This concludes the proof of Theorem 6.1.

88 G. R. Goldstein

The proof in the case of linear growth in (6.4) is more difficult. For details see [17]. The above results can also be extended to include inhomogeneous periodic and nonlinear boundary conditions. The quasilinear version of problem (6.2) has been studied in [12], [13], [17], [30] for various types of boundary conditions. In these papers questions of regularity for both the semi linear and the addressed quasilinear problem. However, in the quasilinear case only local existence is known.

The case of more rapid degeneracy, that is 'Po ( x) -+ 0 arbitrarily fast as x approaches the boundary has been studied by [21] and [31].

Porous Medium Equation The equation

Ut = .6.( uo+ 1 ),

or more generally the equation

Ut = .6.('P(u))

on n c IRn , (6.13)

(6.14)

governs the flow of a fluid throught a porous medium and can be derived under certain approximations from the Navier-Stokes equations. Here Q > O,u(t,x) = cp(t,x) where c > 0 and p(t,x) represents the density of the fluid at the point x E n and at time t E m+, the pressure is given by p(t, x) = Po[p(t,x)]O' for some Po > O. In (6.14) we asume that the mapping 'P : IR -+ IR is either a continuous nondecreasing function or a maximal monotone graph satisfying 0 E 'P(O). Equation (6.13) may be viewed as a special case of (6.12) where

'P(s) = { SO'+l

-lsIO'+l for s < O.

for s ~ 0

We consider the case n = IR n, and the operator A acting on X = LP (IR n) where A is the nonlinear operator

A=.6.0'P

with domain

D(A) = {u EX: there exists w E L1Zoc(IRn)

such that w( x) E 'P( u( x)) and

.6.w EX}.

Recall that in Ll(1Rn), ¢ E J(u) iff 11<1>11 = 1 and < u, >= Ilulll; in particular ¢( u) E sgn u where

and

sgn u = {w E VX)(JRn ): w(x) E sgn (u(x)) a.e. },

sgn s = { ~L 1] { -I}

if s > 0 if s = 0 if s < O.

In addition, we note that

Au = {~w : 1O(:r) E <p(u(x)) for some u E V(A)};

hence, A is single-valued if <p is continuous. The next theorem is the main result of this section. 'Ve break

its proof into several parts.

Theorem 6.2: The operator A is m-dissipative on LP(JRn) iff p=1.

A similar result is valid in Hilbert space context related to H-l(fl).

Lemma 2: A is dissipative on Ll(JRn).

In order to prove Lemma 2, we need the following claim.

Claim: Let Hi E V(A) and AVi = ~1Oi for i = 1,2. There exists ¢ E J(u - U2) n J(WI - W2)'

Proof of Claim: ·Without loss of generality we assume that Ilul - u211l = 1. Since U2(X) implies Wl(X) ~ W2(X) and Ul(X) < U2(X) implies Wl(X) :S W2(X). Whence, if we define ((UI - U2)(X)) = sgn ((Ul - U2)(X)), then = ±1 E sgn ((WI - W2)(X)) as long as Wl(X) -I- W2(X). If x is such that

Wl(X) = w2(:r), let

90

1

o

-1 ifuI(x) <U2(X),

Then this </J satisfies </J E J(WI - W2) n J(UI - U2).

G. R. Goldstein

Proof of Lemma 2: Let Uj E D(A) and .6.Wj = AUj for i = 1,2. By the claim we can choose </J E J( UI - U2) n J( WI - W2). Then (AUI - AU2, </J) = (.6.WI - .6.w2, </J) ::; 0 since </J E J( WI - W2) and .6. is dissipative on Ll (cf. [18]).

In order to complete the proof of Theorem 6.2, it remains to show the range condition; that is we must solve U - ,\Att :1 h for any h E LI (JRn ). We shall actually only prove that A is essentially m-dissipative here. In particular we shall solve U - '\Au :1 h for h E Co(JR n) and for some ,\ > 0 sufficiently large ( and hence for all ,\ > 0).

Set v = 'P(u) or v E 'P(u)

for v E LZoc(JRn). Define 13 : JR -+ JR by f3(x) = t'P-I(x). Clearly,

'P is a maximal monotone graph satisfying 0 E 'P(O) iff 'P- I = '\13 is maximal monotone and satisfies 0 E 13(0). In addition we see that solving U - '\Au :1 h is equivalent to solving

-.6.v + f3(v) :1 9 (6.15)

for 9 E Co(JRn ). vVe briefly describe the method used to solve (6.15). Let 13m be the Yosida approximation of 13 more precisely

1 -1 13m = m(I - (I + -(3) ). m

It follows that 13m is single-valued, globally Lipschitzian and f3m(O) = O. Also,

(1 - '\Pm)-I -+ (I - '\(3)-1 as m -+ 00 (6.16)

for all ,\ > O. Hence, by the Corollary to Theorem 5.3., .6. - 13m is m-dissipative on LI(ffin).

So we have shown that for all E > 0 there exists '11m f; which , satisfies

(6.17)

The main strategy is to let m ---+ 00 and c ---+ 0, but the order in which we do this is very important. We shall show there exists a subsequence of {um,e} which converges to limit v which satisfies (6.15).

Fix c > O. Multiplying (6.17) by sgnoum,e and integrating over IR n, we see that

cllum,elh + J (lm(um,e) sgno(um,e)

:::; (g, sgnoum,e) :::; IIglh

where we have used the dissipativity and the assumptions on (lm. Again, using the equation, we get

(6.18)

Next, multiplying (6.17) by l(lm(um,e)IP-l sgn o ((lm(Um,e)) for 1 :::; p < 00, integrating over IRn , and using Holder's inequality and the monotonicity of (lm, we obtain

Since ~ - (lm is m-dissipative on LP(IRn) for 1 :::; p < 00,

cllum,ellp:::; Ilgllp; whence

(6.19)

(6.20)

It follows that {Um,e} is a bounded sequence in W 2,P(IRn) for 1 < p < 00. By the local compactness of the imbedding of W 2 ,P(IRn) into LT(IRn) if 7' is sufficiently large, we obtain a subsequence, which we again denote by {um,e} such that Um,e ---+ Ve E

WZ2 ,P(IRn), Ve E W 2 ,p(mn ) and lie satisfies oc

(6.21)

Letting c ---+ 0 is more difficult. First we note that if v satisfies -~v + (l(v):3 h, then clearly U E (l(v) should be in Ll(IRn), but in which space should v exist? Take for example the case (l == O. From Newtonian potential theory we know that if v satisfies -~v = h for h E Ll(mn) then

1 hey) vex) = en 1 1 _2 dy , rnn x_yn (6.22)

92 G. R. Goldstein

that is v = En * h where En = Cn I . 12- n. Here n ~ 3, which we assume. We would like to use a convolution inequality like Young's inequality to say that Ilvllp ::; IIEnllpllhllt, but En ~ LP(IRn) for any p E [1,00]. However, En is in the Marcinkiewicz (or weak LP) space L~k(IRn), with p = n~2 which consists of all measurable functions u for which Illulllp < 00. Here

Illulllp = inf{I< > 0: { lu(x)ldx ::; I<lnll where ~ + ~ = 1 in p q and for all measurable Borel sets nof finite measure}.

L~k(IRn) equipped with the norm 111·lllp is a Banach space. One can show that if WI E L~k(IRn) and W2 E LI(IRn), then

(6.23)

--11- --11-

Since En E L~k2 (IRn), it follows the v E L~k2 (IRn) if v solves -~v = h. In addition one can show that if u E L~'k(IR) for some 1 ::; p < 00 and ~U E L 1(IRn ), then

1 U = Cn I . In-2 * ( -~u),

(6.24)

for some constants An, Bn independent of u. In order to let E -t a in (6.21), one must prove that {,Be (ue)}

is a precompact sequence in LIZ (IRn), where oc

,Be = ,B + El,

and that II,Be(ue)lh < Ilgllt. Then ,Be(ue) -t W E LIZ (IRn) and a.e., and by Fatou's Lemma oc

Nonlinear Semigroups

so w E Ll(IRn). Combining (6.20) and (6.24), we see that

Illuelll n~2 ~ Anll~uelll ~ Anllglh

III~ueill n~1 ~ Bnll~uelll ~ Bnllglll

93

and that {u e}, {~ue} are precompact in LIZ (JRn). Hence, using oc Fatou's lemma, we obtain a subsequence { uen } such that U en -+ v

n

in LIZoc(JRn) and a.e., v E L~k2 (JRn), v E Wli~c (JRn ), ~v E n

L~kl (JRn),w = (3(v) and v solves

-~v + (3(v) 3 g. (6.25)

It follows from a form of the weak maximum principle that v is the unique solution of (6.25). This concludes the proof of Theorem 6.2.

We conclude this section with a brief discussion of some of the qualitative properties of the solution of (6.14).

We have shown that our operator A is m-dissipative on Ll(JRn ), so A generates a semi group T(t) via the Crandall-Liggett theorem. The semigroup T(t) satisfies the conservation of mass property if

{ (T(t)f)(x)dx = ( f(x)dx JIRn JIRn (6.26)

for all f E Ll(JRn). Formally, the porous medium semi group satisfies (6.26) since

dd f (T(t)f)(.r)dx = f Ut(t,x)dx tJIRn JIRn

= { ~(<p(u(t,x)))dx=O JIRn by the divergence theorem. The following theorem of Crandall and Tartar [10] shows that the conservation property implies the positivity of the semigroup for contraction semigroups on Ll spaces.

Proposition 6.3: Let X = L 1(n,/-l) where (n,/-l) is a measure space; Zet C ~ X such that if f, 9 E C, then f 1\ 9 E C. Suppose S : C -+ X satisfies

(6.27)

94

for all f E C. Then the following are equivalent:

(i) If g E C, and f ::; 9 a.e., then Sf::; Sg a.e.

(ii) IISf - Sglll ::; IIf - glh for all f,g E C.

G. R. Goldstein

\i\Te shall give the proof that (ii) implies (i), since this is the part we will use. Consider f, 9 E C such that f ::; 9 a.e. Then

210 (Sf - Sg)+dl1 = 10 ISf - Sgldl1 + 10 (Sf - Sg)dl1

::; in If - gldl1 + 10 (J - g)dl1

by (ii) and (6.27). Hence,

210 (Sf - Sg)+dl1::; 210 (J - g)+dl1 = 0

that is Sf ::; Sg a.e.

Again let us consider the porous medium semigroup Tet) on L 1(ffin). By Proposition 6.3, for all t 2: 0 we have

fl ::; h a.e. implies (6.28)

Ul(t,X) = T(t)JI ::; T(t)h = U2(t,X) a.e. and for all t > 0

where Ui(O, x) = fie x) for i = 1,2. N ow let us specialize to the case

t > 0,0' > l,x E ffin. (6.29)

Here n E IN ; also we consider uO'+1 = lulO'u, the odd extension of uO'+l or consider only nonnegative solutions of (6.29). The Barenblatt solution of (6.29) is

where AI > 0 is given (j = _1_"V = 1-28 b = -2.L and a is I 2+O'n, I 0' 2( 0'+ 1)

uniquely determined by

r U(t, :1:; ]\11) = AI for all t > O. JlRn


Note that if Ixl ~ (%)1/2t D, then U(t,x;lI1) = 0, that IS,

U(t, x; M) has compact support. Consider a function u. which satisfies (6.29) with

where 0::; u.o(x)::; U(to,x;Mo)

for some to, Mo and for all x. Then using (6.28), we see that

0::; u(t,x)::; U(to +t,x;Mo)

for all t > 0 and x E IRn , that is,

supp(u(t, .)) ~ BR

where R = (%)1/2 (t + to)D. In particular the solution u.(t,x) doesn't become everywhere positive immediately because the support is contained in some ball. Hence, we have shown that the solution of the porous medium equation has a finite speed of propagation. Contrast this result with the limiting case a = 0, the linear heat equation. \Ve have

1 -lx-yI2 u(t,x) = (47rt)-n/2 e 4t u.o(y)dy

IRn

where u(O,x) = uo(x). (If a = 0, the Barenblatt solution fails.) !f0::; uo(x) (and Uo =I=- 0 a.e.), then u(t,x) > 0 for all t > 0, and x E IRn; that is, the linear heat equation has an infinite speed of propagation.

References

1. D. G. Aronson, The porous medium equation, Nonlinear Diffusion Problems, Lecture Notes in Math. 1229, Springer Verlag, Berlin, (1986), 1-46.

2. D. G. Aronson and Ph. Benilan, Regularite des solutions de l'equations des milieux poreaux dans Rn, C.R. Acad. Sc. Paris, 288 (1979), 103-105.

96 G. R. Goldstein

3. D. G. Aronson, L.A. Caffarelli, and J. L. Vazquez, Interface with a corner point in one-dimensional porous medium flow, Comm. Pure and Appl. Math., 38 (1985), 375-404.

4. D. G. Aronson, M. G. Crandall, and L. A. Peltier, Stabilizations of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal TMA, 6 (1982), 1001-1033.

5. Ph. Benilan, Equations d'Evolution dans un Espace de Banach Quelconque et Applications, Ph.D. Thesis, Univ. of Paris (1972).

6. Ph. Benilan, H. Brezis, and M.G. Crandall, A semilinear elliptic equation in Ll(RN), Ann. Scuola Norm. Sup. Pisa, 2 (1975),523-555.

7. Ph. Benilan, and :M. G. Crandall, The continuous dependence on 'P of solutions of Ut - ~'P( u) = 0, Indiana Univ. Math. J., 30 (1981), 161-177.

8. M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298.

9. M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, brael J. Math, 11 (1972), 67-94.

10. M. G. Crandall and 1. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc. 78 (1980), 385-390.

11. J. R. Dorroh and G. R. Goldstein, Existence and regularity of solutions of singular parabolic problems, in preparation.

12. J. R. Dorroh and G. R. Goldstein, A singular quasilinear parabolic problem in n dimensions, in preparat.ion.

13. J. R. Dorroh and G. R. Rieder, A singular quasilinear parabolic problem in one space dimension, J. Diff. Equations, 91 (1991), 1-23.

14. L. C. Evans, "Nonlinear Evolution Equations in an Arbitrary Banach Space," Math. Res. Center Tech Summary Report No. 1568, Madison (August 1975).

15. L. C. Evans, Nonlinear evolution equations in Banach spaces, Israel J. Math., 2G (1977), 1-42.


16 D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Springer Verlag, Berlin, 1983.

17. G. R. Goldstein, Nonlinear singular diffusion with nonlinear boundary conditions, Math. Methods in the Appl. Sciences, 15 (1993), 1-20.

18. J. A. Goldstein,Semigroups of Linear Operators and Applications, Oxford University Press, New York and Oxford, 1985.

19. J. A. Goldstein, Semigroups of Nonlinear Operators and Applications, in preparation.

20. J. A. Goldstein and C. Y. Lin, Singular nonlinear parabolic boundary value problems in one space dimension, J. Diff. Equations 68 (1987), 429-443.

21. J. A. Goldstein and C.-Y. Lin, Highly degenerate parabolic boundary value problems, Diff. Int. Eqns. 2 (1989),216-227.

22. J. A. Goldstein and C.-Y. Lin, An LP-semigroup approach to degenerate parabolic bundary value problems, Ann. Mat. Pura Appl., 159 (1991), 211-227.

23. E. Hille, Functional Analysis and Semi-groups, Amer. Math. Soc. ColI. Publ. Vol. 31, New York, 1948.

24. E. Hille and R. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. ColI. Publ. Vol. 31, Providence, R. I., 1957.

25. Y. Kobayashi, Difference approximations of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Math. Soc. Japan 27 (1975), 640-655.

26. S. N. Kruzkov, First order quasi linear equations in several independent variables, Math USSR Sbornik 10 (1970), 217-243.

27. C.-Y. Lin, Degenerate nonlinear parabolic boundary value problems, Nonlinear Anal TMA 13 (1989) , 1303-1315.

28. R. Nagel, One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer-Verlag, Berlin, 1986.

98 G. R. Goldstein

29. A Pazy, Semigroups of Linear Operators and Applications to partial Differential Equations,Springer, New York, 1983.

30. G. R. Rieder, Spatially degenerate diffusion with periodic-like boundary conditions, in Differential Equations with Applications in Biology, Physics and Engineering (ed. by J. Goldstein, F. Kappel and VV. Schappacher), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York (1991), 301-312.

31. A. D. Wentzel, On boundary conditions for multidemensional diffusion processes, Theory Prob. Appl., 4 (1959), 164-177.

32. K. Yosida, On the differentiability and the representation of one-parameter semi-groups of linear operators, J. Math. Soc. Japan 1 (1948), 15-21.

A bifurcation problem for point interactions in L2(JR3)

Wim Caspers and Philippe Clement

1 Introduction

Consider the equation (1.1 )

in IR? with 1 < s < ~ and U E L2(IR?). It is known [8] that this problem has no positive solution for>. E IR. In this paper we investigate a slight modification of this nonlinear eigenvalue problem, where we replace the operator -.6. with domain H2,2(IR3 ), by a so-called point interaction centered at zero. Such point interactions (see [1]) can be described as selfadjoint extensions in L2(IR3 ) of the operator -t::, restricted to C8"(IR3\{O}), smooth functions with compact support in IR3 \ {OJ, or equivalently the operator -.6. restricted to H~,2(IR3\ {O}), that is the space of H2,2-functions that are zero in the origin. All selfadjoint extensions are given by a family {-!::J. 1' }1'E[O,211"),

see (2.4) below. These selfadjoint extensions can also be described by means of Fermi pseudopotentials [1], [5].

One interesting property of point interactions is the existence of a positive eigenfunction (also called ground state), for certain values of r, corresponding to a negative eigenvalue >'0 of -/),,1'. This positive eigenfunction is singular at the origin and behaves like ~ there. It is natural to consider the associated problem

99


(1.2)

100 W. Caspers and P. Clement

for these values of T. Theorem 5 below, states the existence of nontrivial positive solutions of (1.1). These solutions are continuous on JR3, except at the origin, where they have a singularity of order ~. Multiplying (1.2) by a test function v E ego (JR3\ {O}) and using the selfadjointness of -.6. T we get

J -u.6.vdx + J lul s - l uvdx = J Auvdx. (1.3) R3 R3 R3

So in this way we obtain singular solutions of the elliptic semilinear problem (1.1) in JR3\{0}. By considering point interactions in U(JR3) for % < p < 3 (see remark following Theorem 5) one can even allow 1 < s < 2. This approach to singular solutions seems to be new. In Theorem 5 we use a bifurcation and a continuation argument to prove the existence of a curve of positive and negative solutions, parametrized by A between the eigenvalue Ao (depending on T) and O.

Note that these solutions are radially symmetric. The rescaled function

w(y) = (-/\)l~su(hY). (A < 0) satisfies the equation

.)

-w"(r) - :'w'(r) + Iw(rW-1w(r) = w(r), r > O. r

This paper also describes the situation where point interactions associated with -.6. are replaced by point interactions associated with -.6. + r, with a potential V belonging to some class C defined below. Full details and complete proofs will appear elsewhere.

2 Point interactions

Usually. see e.g. [1], point interactions (one center, centered at zero) are defined in L 2 (JR3) as selfadjoint extensions of the operator

-.6. : C~ ( m3 \ {O }) ~ L 2 ( JR3) -+ L 2 ( m3)

where C~( JR3\ {O}) denotes the space of infinitely many times differentiable functions with compact support which does not contain the origin. It is well known that all selfadjoint extensions of this restricted Laplacian operator are given by the family of operators {-.6. T }TE(-1l'.1l'] with

D(-.6.T) = {uo+c('P++eiTy_) : uoEH2,2, uo(O) =0, cEct}

-.6. T (uo + c('P+ + eiT y _)) = -.6.uo + c(iy+ - ie iT y _), (2.4)

A Bifurcation Problem 101

(-tV2±tV2·)lz l where 'P± = e 4".lxl and satisfy (=t=i - ~)-2,2'P± = 80 in the sense of distributions. Here 80 denotes the Dirac measure at zero. (Observe that -~". is the usual Laplacian with D(-~".) = H2,2(IR3 ).)

It can be shown (see [1] Theorem 1.1.4) that the essential spectrum of -~T equals [0,00) for every T E (-7r,1T-]' If T E [~7r,7r], the point spectrum is empty. Otherwise -~T has exactly one negative simple eigenvalue

'J

( cOS(l"'+lT))~ . h .. . f· e-~Izl A AO = - 4 1 2 WIt a POSItIve elgen unctIOn 'P-,\o = 4 I I . s we cos 2''T 7r x

pointed out in the introduction, we are interested in the case -~T has a negative eigenvalue. So we assume T E (-7r, ~7r) in the sequel.

Using the so-called Fermi pseudopotential (see [1], [5]), point interactions can also be denoted as follows. First we consider the closure of -~ in H-2,2(IR3 )

This operator will be perturbed by an extended Dirac measure, which is not only defined on the (continuous) elements of H 2,2(IR3 ), but also on :p+ and 'P-. We define this extension in the following way

471"8 :/ : H 2•2(IR3 ) e ['P+ + eiT <p-l ---t H-2·'2(IR3 )

and for U E H 2.2(IR3 ) and c E u;

c a ( iT )) 471"U ar r u + c( 'P+ + e 'P- :=

4" [:r r( U + c{y+ + eiT 'P-))] (0).80 =

[.bU(O) + c( -~v'2 + ~V2i) + ceiT ( -~v'2 - ~v'2i)] .80 •

Definition 1 For 0' E IR the part of -~-2,2 + 471"0'8 ;rr in L2(IR3 ) IS denoted by -~ + 17ro:S5,-r.

By calculation and using the remarks made earlier, one can prove

102 w. Caspers and P. Clement

Proposition 2 Define for T E (-7r, t7r) the coefficient a.,.

Then -~.,. equals -~ + 47ra.,.8;rr on D(-~.,.). Moreover

D(-~.,.)={UO+C'PI : uoEH2,2, Uo(O) =0, cE(£'} ~

COS ~'1' cos( t1l"+~"') .

-i,:-Izl . . . .. 1 where 'P ~ = e 411"Ixl IS an eIgenfunctIOn of -~.,. wIth eIgenvalue - a~ .

(Note th":t T --t a.,. defines a bijection from (-7r, t7r) onto (0,00 ).)

Consider the following class of potentials C

C:={V=Vi+V2EL2 +Loo : V;:::O, lim V2(x) =0, ~VEL;oJ. Ixl-->oo r

As H 2,2(JR3) can be imbedded into the space of continuous functions and 'P:!T has a singularity of order ~, the perturbation of -~ + 47ra8 ;r r by V is

w;ll defined as an operator in L2(JR3). It can be shown that for every a E (0,00) the operator (-~ + -±lra8 ;r r) + V can also be obtained as selfadjoint extensions of -~ + V, restricted to CO'(JR3 \{0}). Therefore we shall write -D. + V + -±7ra8;rr instead of (-~ + V) + 47ra8;rr and call it a point interaction. In the following theorem we summarize some properties of point interactions.

Theorem 3 (1) Let a > o. The essential spectrum of the point interaction -D.+47ra8 ;r r is [0, ex::). There is exactly one eigenvalue - ;2' which is simple,

with corresponding eigenfunction 'P ';2 = e;1I"~~~I. Moreover 'P ~ minimizes the functional

under the condition 11/112 = II'? 1 Ib-;;2

(ii) Let a > 0 and F E C. The essential spectrum of the point interaction -D. + F + -±7ra8 ;rr is [0. x). This point interaction has at most one negative eigenvalue AO, which is simple, and !./J>.o := (,\0 - D. + V)::::b80 is an eigenfunction. Tllis eigenfunction minimizes the functional

1 ~ --\ J 1I1 2dx + J V''P~1 f V''P~1 1 'P'i..dx + J VI/1 2dx a 02;;2 02 R3 R3 R3

under the condition II fib = 11~.\o 112' Consequently Ao 2: - ;2' Statement (i) is a consequence of Proposition 2 and the remarks made at

the beginning of this section. See also [3] Chapters 3 and 9. For the proof of (ii) we refer to [3], Chapters 3 and 5. There, explicit expressions for the resolvents of point interactions are given. The statement on the essential spectrum and the fact that the point interaction has at most one negative eigenvalue, which is simple, can then be derived by using the WeinsteinAronszajn determinant (see e.g. [6] IV 6).

3 The nonlinear problem

Let VEe and a > O. We denote the operator -~ + V + 47rao ;rT in L2(m3) by Ba.. Assume Ao < 0 is an eigenvalue of Ba. with eigenfunction ~.\o' Consider

(3.5)

where 1 < s < ~, A E m and u = Uo + c'P-!z E D(Ba.). (It fo11O\vs from

1 < s < ~ that for u E D(Ba.) we have luls-IU" E p(m3 ).)

Let (X.llllx) denote D(B~) supplied with the norm. defined by Iluo + c(y .,:2 )llx := Il uoil2.2 + lei for Uo + c('P;!2) E D(Ba.) .

Definition 4 A solution (to problem (3.5)) is a pair (,\, u) E m x X satisfying (:3.5). A solution (A, u) is called positive if u > O.

Theorem 5

1. If (,\, u) is a nontrivial solution then A > Ao.

2. If(A, u) is a nontrivial positive solution then A E (Ao,O].

3. There is a constant JI > ° such that: if (A, u) is a positive solution then f u~'.\odx :S jllA - '\01 '~1 and f lu!s+ldx :S AI and f lul 2dx :S ~~.

W W W

4. There is a Cl-function W : Po, 0) ~ X such that (,\, W(A)) IS a positive solution for A E ('\0,0) and W(A) --+ 0 in X as A 1 Ao.

5. For>.. E (>"0,0) the problem (3.5) has exactly three solutions: (>.., \11(>..)), (>..,0) and (>.., -\11(>")).

Remark Point interactions are also well defined in U(IR?) for ~ < p < 3 as negative generators of analytic semigroups (see [2] and [3]). If in (1.2) -~7" is replaced by a point interaction in U(IR?) for suitable ~ .. > >"0. 2. If (>.., u) is a nontrivial positive solution then>.. E (>..0,0).

Suppose (>.., u) E IR x X is a nontrivial (positive) solution of (3.5). Then u is a (positive) eigenfunction with eigenvalue>.. of the operator Bcx + luis-I. Note that lul s-I E C and therefore lui s - I + V E C. So u is a (positive) eigenfunction with eigenvalue ,\ of the operator

and we can apply Theorem 3 to conclude that>.. > - ;2 (>.. E (- ;2,0)). Comparing the quadratic forms of the operators - ~ + V + 471"a8 ir rand -~ + V + lul s-1 + 47r8:r r it can be shown that>.. 2:: >"0 .

.9. There is a constant M > 0 such that: if (>.., u) is a positive solution then J u!/'.\odx :::; I) - '\01 ,.:.\ and J lul s+1dx :::; j\;1 and J lul 2 dx :::; AI. ~ ~ ~

If we multiply (3.5) by W.\o and integrate over IR3 , Jensen's inequality (see e.g. [9]) can be used to derive the desired estimates.

4. There is a CI-function \II: (>"0,0) ~ X such that (>.., \11(,\)) is a positive solution for>" E (>..0, 0) and \II (,\) --t 0 as ,\ 1 Ao.

Define F : IR x X --t r(JR3) by

Lemma 6 The partial derivatives Fu, F>., and F>.,u exist and are continuous. Moreover for (A, u) E m x X and v E X

Fu(A, u)v = BO/v + sluls-1v - AV F>.,(A,U) =-u F>.,u(A, u)v = -v

and N(Fu) is spanned by"p>.,o.

(Note that F is not twice differentiable with respect to u as 1 < s < n By applying Theorem 1.7 of [4] one can prove

Proposition 7 There is a neighbourhood U of(Ao, 0) in mxx, an interval (-E, E) and continuous functions A : (-E, E) --t m and X : (-E, E) --t

H2•2(m3 ) such that A(O) = - ;2' X(O) = 0 and

F-1(0) n U = {(A(S),S("p>.,o + X(s))) : lsi < E} U {(s,O) : (s,O) E U}.

Moreover w( s) := s( "p>"o + X( s)) is positive (resp. negative) when s > 0 (resp. s < 0).

Using the implicit function theorem it can be shown that, given a nontrivial positive solution (A, u), locally all solutions can be described by a C1-curve W(A). This is stated in

Proposition 8 Let (A, u) E (Ao,O) x X be a nontrivial solution. Then there is a neighbourhood N of A in m and a Cl-curve W : N ---t X such that W(A) = u and for J1 EN: (J1, W(J1)) is a solution of (3.5). Moreover there is a neighbourhood U of (.>., u) such that all solutions of (3.5) in U are of the form (J1, W(J1)).

So far we have proven the existence of a C1-curve W : ('>'0'5.) ---t X where 5. E (Ao,O], such that (A, W(A)) is a positive solution for A E (Ao,5.) and W(.>.) ---t 0 as A ---t '>'0. It is a consequence of Proposition 8 and the following proposition that 5. can be taken equal to O.

Proposition 9 Either liID W(A) exists in X and (~, liID W(A)) is a solution, >.,p >.,p

or 5. = o.

106 w. Caspers and P. Clement

We give an outline of the proof. Suppose ~ < O. Let An be a sequence in ('\0,0) such that An i ~ as n ---t DC, and define Un := Il1(An). It can be shown that Un! ::; Un2 if nl ::; n2 so by 2. and Beppo Levi's Theorem

for u( x) := sup Un (x). The latter statement can be rewritten as nEN

I Is I-Is. L1+1 Un ---t U In s •

Note that 1 + ~ < 2 and 1 + ~ > 1 + ~ = ~ > ~. For w ~ 0

WUn + Baun = -Iunl s + (An + w)un.

In [2], [3] it is shown that for W large enough (w + Batl is bounded on L1+~(JR3) and on L2(JR3). So writing UI,n := (w + Ba)-l( -lunIS) and U2,n := (w + Batl(Pn + w)un) and UI .- (w + Batl( _luiS) and U2 :=

(w + Batl((,\n + w)u) we have

and

Using this. one can show that, writing

Un = UO.n + en 1/;'\0

and u = Uo + C"lJ),\o

with UO,n E H2,2(JR3), iio E H2,2(JR3) + H2,1+~(JR3) and en, C E (£', that lIuo,n - uoll oo ---t 0 and len - ci ---t o. From this we deduce that u E D(Ba) and Ilun - ul/x ---t O.

5. For A E ('\0,0)) (3.5) has exactly three solutions (,\, Il1P)), P,O) and (A, -1l1(A)).

So far we have established the existence of a Cl-curve III : (/\0,0) ---t X such that (A, Il1(A)) is a solution for all A E (AO,O) and Il1(A) ---t 0 as A ---t Ao and

'l1(>') > O. Clearly also (>.,-w(>')) and (>.,0) are solutions for>. E (>'0,0). Suppose (/L, v) is a nontrivial solution for some /L E (>'0,0). Without loss of generality we assume v > O. We will show that v = W(/L). It follows from Proposition 8 and Proposition 9 that there is a fl E (>'0, /L) and a C1-curve <1> : (fl,O) -t X such that (>., <1>(>')) is a solution for all>. E (fl,O) and <1>(/L) = v and <1>(>') > O. Following the proof of Proposition 9 (in fact some steps can be simplified in this case) it can be shown that <1>(11) := 11m- <1>( ,\) exists and

(fl, <1>(11)) is a solution. So either <1>(fl) > 0 or <1>(11) = O. If <1>(fl) > 0, the curve <1> can be continued by Proposition 8 for /L < fl in some neighbourhood of fl. If <1>(fl) = 0 then fl = >'0 as >'0 is the only point of bifurcation by Proposition 7. From Proposition 7 it follows that the curve <1> and Ware the same and consequently W (/L) = v.

References

[1] S. Albeverio, F. Gesztesy, R. H0egh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, Springer, New York, 1988.

[2] W. Caspers, Ph. Clement, Point interactions in U, Delft report 91-97, 1991, to appear inSemigroup Forum.

[3] W. Caspers, On Point Interactions, thesis in progress, Delft University of Technology.

[4] M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Anal. 8 (1971), 321-340.

[5] E. Fermi, Sui moto dei neutroni nelle sostanze idrogenate, Ricerca Scientifica 7 (1936), 13-52. English translation in 'E. Fermi Collected papers', Vol. L Italy, 1921-19:38. Cniversity of Chicago Pres, Chicago-London, 1962, 980-1016.

[6] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin-Heidelberg-New York. 1966.

[7] M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness,Academic Press, 1972.


[8] W.-M. Ni, J. Serrin, Existence and nonexistence theorems for ground states of quasilimear partial differential equations. The anomalous case, University of Minnesota, Mathematics report 84-150.

[9J M. Reed, B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978.

[10] B. Simon, Schrodinger semigroups, Bull.Amer.Math.Soc., 7 (1982), 447-526.

Semipositone problems*

Alfonso Castro and Ratnasingham Shivaji

1. Introduction.

By a semiposicone problem we mean a semilinear equation where the nonlinearity IS

nondecreasing and negative at the origin. A typical example is the Dirichlet problem

6u+,\f(u) =0 in rt, u=00n8rt (Ll)

where ,\ E (0, (0) is a parameter, rt is a smooth bounded region in Rn, 6 is the Laplaciall

operator and f : lR -+ lR is a locally Lipschitzian monotonically increasing function such

that

f(O) < 0

ane. flu) > a for some u > O. Se::cipositone problems nabrally arise in various stuc.:es.

For exa..'1lple, cor.sider the RozeIlwig-).1cArtb.:r equations in the analysis of compec:!lg

species where "har-;esting" ta..1.;:es place (see[l]). The study of positive solutiocs to

subject to (1.2), lli-ilike the positone case (J(O) ;:: 0), tu:-ns into a nontrivial question

as rP == 0 is not a subsolution, making tce r::ethod of sub-super solutions ddicu..:t co

apply. Sewipositone problems, again unlike positone problems, give rise to the interesting

phenomenon of sy=etry breakillg. The reader is referred to the work by Smolle:- 8.:

vVa5serman (see [2]) for seminal work in this di=ection. The reader is refer=ed to the ~vo:-':

by Ar::L."::J.an (see [3]), Laetch (see [..1,; KeUe>Cohen (see [.5];, Brovm-Ibr2.hi::..,-Shivaji

'Supported in par: by ",SF Grant D\,[S - 8905936

109

110 A. Castro and R. Shivaji

The rest of the paper is divided into six sections: positive solutions when n = 1, positive

radial solutions, sign changing solutions when n = 1, sign changing radial solutions, positive

solutions in general bounded regions, and open problems.

2. Positive Solutions when n = 1.

Noting that solutions to

u" + )..j(u) = 0 (2.1)

are symmetric about its critical points, i.e., if u' (a) = 0 then u( a + x) = u( a - x), and

satisfy

u' = ±(C - 2,\F(u))~ where F(s) = 1s /(t)dt. (2.2)

the following results can be proven. Let /3, e be the unique positive zeros of f, F respec

tively. and let A = 2Uoe(-F(s))-~dsJ2. Then:

Theorem 2.1. Let / be convex and superlinear. Then the boundary value problem

u" + A/(u) = 0; u(O) = 0 = u(l) has a unique positive solution for A E (0, .\] and no

positive solutions for .\ > .\. Further, there exists a unique non-negative solution with

(n -1) interior zeros at ,\ = n21\ for n E N.

Theorem 2.2. Let f be concave, lims_x f(s) = jf.O < lVI S; oc and lims_ oo .sf(5) = O.

Then there exists a K E (0 .. \) such that the boundary value problem u" + A/(u) = 0; u(O) = 0 = u(1) has no positive solutions for ,\ E (0. K), exactly two positive solutions

for A E (1\, AJ, and a unique positive solution for)' E {K} U (A. (0). Further. there exists

a unique non-negative solution with (n - 1) interior zeros at A = n 2 A for n E N.

We refer the reader to [7] for the proofs of these above results, further details, and for

discussion in the case of concave-convex type nonlinearities.

Semipositone Problems III

Theorem 2.3. Assume that the set S = (7r2 n2 j f'((3), (J2 j[-2F(}1)]) is non-empty where

n EN. Then the boundary value problem u" + Af( u) = 0; 'u' (0) = 0 = u' (1) has at least

2n + 1 positive solutions for each A E S.

See [8] for the proof of Theorem 2.3 and further details.

Theorem 2.4. Let f be convex and superlinear. Then there exists f(a) E (0, A) such

that the boundary value problem u" + Af(u) = 0; u(O) = 0 = u(l) + au'(1), where

a E (0, 00), has a positive solution for each A E (0, f( a)], and no positive solutions for

A > A. Further, there exists fn(a) E ([2n -1]2Aj4,n2A) such that there exists unique

non-negative solutions with (n - 1) interior zeros at A = f n( a) and at A = n2 A for n EN.

For the proof of Theorem 2.4, and further details including the evolution of these solu

tions as Q varies, see [9].

We close this section by also referring to [10], where the evolution of the branch of

positive solution for the boundary value problem u" + Af( u) = 0; u(O) = 0 = u( 1) as the

nonlinearity f evolves from having one positive zero to three positive zeros is discussed.

3. Radial Positive Solutions.

l'nlike in the one dimensional case. when f2 is a ball in Rn(n > 1), the problem (1.1) with

Dirichlet boundary conditions has no non-negative solutions with interior zeros. Indeed

in [ll] the authors proyed the following result which answered an open question posed in

the celebrated paper of Gidas - :.'i'i - ~irenberg (see [12]).

Theorem 3.1.

Let n be a ball in Rn(n > 1) and u ~ 0 be a solution of the boundary value problem

D.u + Af( u) = 0: x E II, II = 0; .r E all. Then 11 > 0 in II and hence, radially symmetric.

On the structure of positive branch of radial solutions to the boundary value problem

D.u + Af( u) = 0: x E II, II = 0; x E oil when II is a ball in ]Rn( n > 1) the following results

are known.

Theorem 3.2. Let f be convex, superlinear. and subcritical (i.e. there exists A and

p E (O,[n +- 2]/[n ~ 2]) such that Iflu)l::; A(l + uP). Then there exists a unique positive

solution for A small and no positive solution for ,\ large.

The reader is referred to [13]-[16] for the proofs of these results.

Theorem 3.3. Let f be cOJ]cave and 5ublinear. Then there exists 6] < 62 such that for

/\ E (0. 6ll there are no posi ti,'e ,olu tioJ]s. for ,\ E (6] ,x;) there exis,s a unique stable

positive mllltioll. <lnd for ,\ E [,11. there exists a unique unstable positive solution.

For the proof of these results sec '17]-[18].

4. Sign Changing Solutions When n = 1.

Like in tl1(" case of positi,'e solutions when n = 1. USll1g the quadrature method a

complete study of sign changing :iOlutions can be achieved. In particular. see [19] where

the authors establish:

Theorem 4.1. Let f be supeI1inear. f( .0) convex for s > 0 and concave for s < 0, and

lims--x(f(s) ~ sf'(s)) > O. Then for each A E (0. n2 A) the boundary value problem

u" + Af(u) = 0: u(O) = 0 = u(1) has at least one solution u with 2n interior zeros with

u'(O) < 0, at least one solution u with 2n ~ 1 interior zeros with 11'(0) < 0, at least one

Semipositone Problems 113

solution u with 2n - 2 interior zeros with u'(O) > 0, and at least one solution with 2n - I

interior zeros with u'(O) > 0, Further there exists a A* E (0, A] independent ofn such that

for A E (0, A*) the above type of solutions are unique.

Also see [20] where sign changing solutions for the boundary value problem u" +Af( u) =

O;u(O) = 0 = u(l) + au'(l) are discussed in detail. In particular the following existence

result is proven.

Theorem 4.2.

Let f be super linear. Then there exists two sequences of nontrivial bifurcarion points

{(An(a), B)}:;O=l' {((Tn' B)}~=l with An(a) < an and B > 0 is such that F(B) = 0, Iv'here

F(s) = fa' f(t)dt,for the boundary value problem u" +Af( u) = 0, u(O) = 0 = u( I )+au'( I),

a E (0, Xl). Further the branch of solutions (A, u) where u has (2n - 2) interior zeros and

u'(O) > 0 and the branch of solution:; (A, 11) .vhere 11 has (2n - 1) interior zeros and

u' (0) < 0 bifurcates from (A n (a), B), while the branch of solutions (,\, u) where II has (2n)

interior zeros and tI'( 0) < 0 and the branch of solutions (A, u) where u has (2n - I) interior

zeros and u'(O) > 0 bifurcates from (an' 0).

See also [20] for the study on the eyolutioll of the above branches as a varies

5. Sign Changing Radial Solutions.

Cnlike positive solutions to (1.1" sign changing solutions in the case when 0. is a ball

are not necessarily radially symmetric. However for radial solutions we have the follow

ing description of its branches. In what follows S denotes a connected component of

{(A, U)IA > 0, u E C(Q), u is radiaL and (A, u) satisfies (U)}. Then we have:

Theorem 5.1.

For each non empty 5 there exists a nonnegative integer k such that if (,\, u) E 5 then

U has either 2k or 2k + 1 nodal hypersurfaces.

Theorem 5.2.

Suppose (AO, uo) E S. The function Uo has 2k nodal hypersurfaces in nand Vuo(x) =f. 0

for x E an iff there exists (AI, UI) E 5 satisfying uo(O) = UI(O) and Ul has 2k + 1 nodal

hypersurfaces in n. Here k is a nonnegative integer.

Theorem 5.3.

For any nonnegative integer k, there exists a unique unbounded branch of solutions

5 = Sk (sa}) such that if (A. tl) E Sk then U has either 2k or 2k + 1 nodal hypersurfaces

in O.

Theorem 5.4.

There exists AI > 0 such that H(A, u) is a solution to (1.1) with A < M, then (,\, u) E Sk

for some nonnegative integer k. j\'{oreover Vu(x) =f. 0 for x E ao.

Theorem 5.5.

If f is convex and f(t)/[tf'(t) - f(t)] is a non decreasing function then for each A > 0

the problem (1.1) has at most one positive solution u such that (A, u) E So.

The reader is referred to [15] for proofs of Theorems 5.1-5.5 and further details.

Semipositone Problems liS

6. Positive Solutions in General Bounded Regions.

Suppose now that D is a smooth bounded region without any special symmetry. Then

the following results are known to date.

Theorem 6.1.

Let f be a superlinear and a.ssume that there exists p E (1. [n + 2J/[n - 2]) and real

numbers A and B such that A(lulP - 1) ::; f(u) ::; B(lulP + 1). Then there exists (J > 0

such that for A < (J the problem (1.1) has a positive solution.

This and related results have been proven in [21]-[22]. An array of methods in nonlinear

functional analysis such as the mOllntain pass lemma. Leray _. Schauder degree. and a priori

estimates for positive solutions were ingeniously utilized by the authors in pro\'ing these

results.

Next in [23] the aut hms by nontrival usage of Green's identities prove the following

theorem. which also holds in the ca:;c of Robin boundary conditions.

Theorem 6.2.

If f is convex then e\'ery non-negative solution of (1.1) is unstable.

The study in the case when f i:; concave has been achieved via the sub-super solu

tion method. In particular, in [2], [24] and [25] the authors succeed in producing the

much needed non-negati\'e subsolution. In fact, in certain ranges of ,\, an anti-maximum

principle by Clement and Peletier (see [26]) was used in establishing this non-negative

subsolution. The summary of the existence results in the f concave case is as follows:

Theorem 6.3. If f is concave there exists L < T such that for A > T the problem (1.1)

has a positive solution, and no positive solutions for A < L. Further for classes of concave

nonlinearities (1.1) has a positive solution close to the smallest eigen value of -6 with

Dirichlet boundary condition.

For related results in the case of Robin boundary conditions see [27J.

7. Open Problems.

It seems that a rather significant analysis has been achieved in the case when (1.1)

reduces to an ordinary differential equation. When this is not the case, many questions

remain unanswered. In this section we summarize some of these important open questions.

(1) Is there a bounded smooth region D in which (1.1) has a nonnegative solution with

interior zeros?

Note that by the maximum principle this is not possible in the case of positone

problems. Also recall that even in the case of semipositone problems this is not

possible when D is a ball in JRn; n > 1. However. note that for semipositone

problems such solutions exists when say D = {(x, y)/O < x < 1, -oc < y < oc}.

This follows easily by recalling that such solutions exists when n = 1.

(2) Given a general bonnded region D, when f is convex and superlinear, it is true Jhat

there exists at most one positive solution? Also when f is concave and sublinear,

is it trne that there is a most one positive solution for A large?

(3) Study of existence of multiple positive solutions for concave-convex type nonlin

earities when n > 1.

See [7J where multiplicity results were discussed in this situation when n = 1.

(4) Study of sign changing solutions for general bounded regions D in Rn; n > 1.


(5) Study of systems of equation" which are of semipositone nature.

REFERENCES

1. 0,1. R. ~Iyerscough. B. F. Gray, W. L. Hogarth, and J. 'lorbury, An analysis of an ordinary differential equations model for a two speCleS predator - prey system with harvesting and stocking. J. 'vlath. BioI. 30 (1992), 389-411.

2, J. Smoller and A. \Vassernlan~ Existence of positive soLutions for semilinear elliptic equations in general domains, Arch. Rational Mech. Anal. 98 (3) (1987), 229-249.

3. H. Amman, Fixed point equations and nonlmear eigenvalue problems in ordered Banach spaces, SIA'v1 Rev. 18 (1976). 620-709.

4. T. \V. Laetch. The number of solutzans of a nonlinear two point boundary value problem. Indiana l.:niv. Math. J(20) (1970/71). 1-U.

,5. H. B. Keller and D. S. Cohen. Sorne posilone pToblerns suggested by nonlinear heat generation . .J. ~Iath. Mech. 16 (1976).1:361-1376.

6. K. J. Brown. 'v1.'vI..-\. Ibrahim. and R. Shi,·aji. S·shaped bifuracation curves. J. :\onlinear Analysis. T'vIA 5(5) (1981).47.')-486.

i. A. Castro and R. Shi\"aji. Nonneg(ltive .wluiions for a class of nonposdone problems. Proe. Roy. Soc. Edin. 108(A) (1988) 291-302.

8. A. ~liciano and R. Shivaji, Mult.ple po.;.twe solutzans for a class of semipositone Neumann two point boundary value problems. J. 'vlath. Anal. .\ppl. (to appear!.

9. V. Anuradha and R. Shivaji. Nonnegatzve s(Jluttons for a class of superlinear multiparameter sem.positone problems. Prcprillt.

10 . .-\ .. Khamayseh and fl. Shi\"aji. Evolutton of bifurcation curves for semipos,ione problems when nonlinearities develop multiple zeroes, J. App!. ~Iath. and Compo (to appear).

11. A. Castro and R. Shivaji, Nonnegative solutions to a sem.linear Dirichlet problem in a ball are posit271e and radially symmetric. Comm. in PDE 14(8 & 9) (1989), 1091-1100.

12. B. Gidas, W. Ni, and L. \Tirenberg, Symmetry and related properties via the maximum principles, Comm. Math. Phys. 68 (1979),209-213.


13. A. Castro and R. Shivaji, Nonnegative solutions for a class of radially symmetric nonpositone problems, Proc. Amer. Math. Soc. 106(3) (1989), 735-740.

14. K. J. Brown, A. Castro, and R. Shivaji, Nonexistence of radially symmetric nonnegative solutions for a class of semipositone problems, J. Diff. & Int. Eqn. 2(4) (1989), 541-545.

15. A. Castro, S. Gadam, and R. Shivaji, Branches of radial solutions for semipositone problems, Preprint.

16. A. Castro and S. Gadam, Nonexistence of bounded branches and uniqueness of positive solutions in semipositone problems, Preprint.

17. I. Ali, A. Castro, and R. Shivaji, Uniqueness and stability of nonnegative solutions for semipositone problems in a ball, Proc. Amer. Math. Soc. (to appear).

18. A. Castro and S. Gadam, Uniqueness of stable and unstable positive solutions for semipositone problems, J. Nonlinear Analysis, TMA (to appear).

19. V. Anuradha and R. Shivaji, Existence of infinitely many nontrivial bifurcation points, Resultat der Mathematik (to appear).

20. v. Anuradha and R. Shivaji, Sign changing solutions for a class of superlinear multiparameter semipositone problems, Preprint.

21. S. Unsurangsie, Existence of a solution for a wave equation and elliptic Dirichlet problems, Ph.D. Thesis(1988), Univ. of North Texas.

22. W. Allegretto, P. Nistri, and P. Zecca, Positive solutions of elliptic nonpositone problems, J. Diff. & Int. Eqns (to appear).

23. K. J. Brown and R. Shivaji, Instability of nonnegative solutions for a class of semipositone problems, Proc. Amer. Math. Soc. 112 (1991), 121-124.

24. K. J. Brown and R. Shivaji, Simple proofs of some results in perturbed bifurcation theory, Proc. Roy. Soc. Edin. 93(A) (1982), 71-82.

25. A. Castro, J. B. Garner and R. Shivaji, Existence results for classes of sub linear semipositone problems, Resultat der Mathematik (to appear).

26. Ph. Clement and L. A. Peletier, An anti-maximum principle for second order elliptic operators, J. Diff. Eqns. 34 (1979), 218-229.

27. W. Allegretto and P. Nistri, Existence and stability for nonpositone elliptic problems, J. Nonlinear


Analysis, TMA (to appear).

UNIVERSITY OF NORTH TEXAS DENTON, TX 76203

MISSISSIPPI STATE UNIVERSITY MISSISSIPPI STATE, MS 39762

A generation result for C-regularized semigroups

IoanaCioranescu

Abstract. In this work we give a generation theorem for C-regularized

semigroups which generalizes Oharu's results of [7]

Let X be a Banach space and C an injective bounded linear operator in X with

dense range. According to Da Prato [4] (see also Davies and Pang [3]) we say

that the family {Stt~o of bounded linear operators in X is a C-regularized

semigroup if the family is strongly continuous and

Exponentially bounded C-regularized semigroups were introduced and

extensively studied in connection with the abstract Cauchy Problem. One of the

most important result of the theory is the following Hille Y osida type theorem:

Let A be a linear, densely defined operator on X S.t. (ro,oo) c p (A), for some

ro ~ 0; then the following are equivalent:

a) A generates a C-reguJarized semi group {St} t~O satifying II St II $ Mewt , for

t ~ 0 and some M > O.

b) C commutes with all the resolvents of A and

(see [3], [4], [5], [6], [9]).

121


122 I. Cioranescu

We note that de Laubenfels proved the above result without the hypothesis

that C have dense range; moreover he initiated in [5] the study of C-regularized

semigroups that may not be exponentially bounded. We present here a generation

result for such C-regularized semi groups which generalizes the following result of

Oharu (Theorem 2.3 of [8]): for a linear densely defined operator A such that

(ro, 00) c P (A), the following condition is sufficient in order to generate a

continuously differentiable semi group of operators on D(Ak):

for every t ~ 0 there is a M(t) > 0 such that for A > ro, 0:5 i :5 t, n E N, x E D( A k)

n

II AnR(A; A)n x ll:5M(t)ll x llk, wherellxllk = IJ Aj4 j=l

Throughout this work C is linear, bounded, injective with dense range C(X), A

has the domain D(A)cX, peA) is the resolvent set of A, R(A; A) the resolvent

function at AE p (A).

Our main result is:

Main Theorem: Let A be a densely defined linear operator satisfying the

conditions:

i) there exists OJ> 0 so that (OJ, 00) cp(A) and R(A; A) C = C R(A; A)for A> OJ;

ii) for every t > 0 there exist a M(t) > 0 such that

Then A generates a C-regularized semigroup which is O(M(t)) as t ~ 00.

Proof: We denote Th = (I - hArl; then the condition ii) can be written as:

for every t > 0 there is a M(t) > 0 satisfying

A Generation Result 123

II CJ~II~ M(t) for h -I> 0) and 0 ~ nh ~ t. (1)

We shall prove that S x = lim C Jhi t/h 1 exists for every x E X, uniformly for t in t h--)O

+

finite intervals of R+, and that {StL;:::o is a C-regularized semi group of

generator A with II St II = O(M(t»). We note that

(2)

Indeed, for x = Cy, Y E D(A), we have

Moreover it is an easy mailer to see that C(D( A)) = X, also using (2) we obtain

(3)

We can now adapt Oharu's method (see [7]) to our case.

Let XE D(A2),tO >0,tE [0, to] and rl~h-I>O);thenwehave

(4)

Let us take f = T j, h = 2-m, n = 2j-m and k = [2mt], where j, mEN and j ~ m;

thenh-nf=Oand khE rO,t01.

124 I. Cioranescu

Using (1) we have r

IIC2J~t/ilx_C2J~kxll=£ LC2J1+PAx :S;toIICIIMIIAxll:S;hIlCII M(tO) II Ax II p=l

where 0 :s; r = [t / £] - nk :s; n.

Since J~x = J~/hl, in order to estimate (4) we only have to calculate

II C2J~kx - C2J~x II. We note that

But we have

k-1 JTIk Jk - ~(Jk-iJTIix Jk-i-1JTI(i+l)x) f x - hX - - L.. h l - hI.'

i=O

As O:s; (k - i) h :s; kh:S; to and O:s; (ni + q + 1)£:S; k£n :S;to we can use (1) to get

k-l n-l n-l

IIC2J~kx - C2J~xll:s; £2LIICJ~-i II L LIICJ~i+q+lA2xll i=O p=l q=p

Then, for x E DCA 2), to > 0, £ = 2-j and h = 2-m, with 2j ~ 2m > ro, we have


It follows that lim C2J~/h]x exists unifonnly for t E [0, to]' Then by (3) we h=2m~0

have that

S x = lim CT[t/h]x t > 0 t h=2-"~O h '

(5)

exists for every x E X, uniformly on finite subintervals of R +. It is also clear that

(6)

We shall prove that {StL,:o defined by (5) is a C-regularized semi group

whose generator is A. Consider x E D(A) and t > 0; then the functions

s ~ CJ~/h] Ax are step functions on [0, t], unifonnly bounded with respect with

SE [0, t] and h = 2-m, mEN.

Since [t/h]-l

Jl t/h lx - x = h "Jk Ax - hAx + hJ[t/h]Ax h ~ h h

k=O

then t t

CJ[t/h]X - Cx = fCJ[s/h]Ax ds- fCJ[S/h}Ax ds- hAx + hT[t/h}Ax h h h h'

o [t/h]h

Letting h = 2-m ~ 0, we obtain

t t

StX - Cx = f SsAx ds = A f Ssx ds. (7) o 0

In particular it follows that t ~ St x is continuous on [0, oc), for every x E D(A);

then the density of D(AJ and (6) yields the strong continuity of the operator

126 I. Cioranescu

family {Stt~o' We can now use Theorem 2.6 in [5] to conclude that {St} t~O is a

C-regularized semigroup generated by an extension A of A. We shall finally

prove that A = A. Indeed, we note that by the closedness of A and (6), we obtain

from (7) t t f Ssx ds E D(A) and SIX - Cx = A f Ssx ds, x E X. (8)

o 0

Let x E D(A); then by (8)

[ t )

S x-Cx A ,-1 [s, x<is : t ,

t

-----t) CAx. t~O .

Since lim t -1 f S xds = Cx and A is closed, we get t~O s

o Cx E D(A) and ACx = CAx, x E D(A).

It follows that CD(A) c D(T) and AICD(A) cA. Since CD(A) is a core of A

(see Tanaka [9], Theorem 2.1) A = AICD(A) c A. Therefore A = A, i.e. A is the

generator of { S t} t~O . We note that one can prove as in [7] that the limit in (9) is independent of the

chosen sequence, so that we have

Corollary 1. Suppose A is as in the Main Theorem; then the C-reguZarized

semigroup generated by A is given by

S x = lim (I - hSr[t/n1x t h~O

(9)

where the convergence is uniform in t on bounded intervals.

We also mention that Tanaka [9] proved that formula (9) is true for every

exponentially bounded C-regularized semi group.

Corollary 2. Let A be a densely defined operator satisfying the conditions:

a) there is OJ> 0 s.t. (OJ, 00) < peA)

b) for every t > 0 there is M(t) > 0 such that

forsomekE N,XE o (Ak), A >O)andO::;"i::;t, nEN.

Then A generates an R(A; Ayk-regularized semigroup which is O(M(t)),for all

A> OJ.

Proof: We only have to take in the above theorem C = R(A; A)k , A > 0).

Remark 1. This result is essentially Theorem 4.2 in Oharu's [7]. We note that

condition b) is also necessary in order that an densely defined operator A with

(0),00) c peA) be the generator of an R(A; A)k-regularized semigroup; see: the

Main Theorem in Sanekata [8] and Corollary 4.5 in de Laubenfels [5]. We can

not prove that condition ii) in our Main Theorem is also sufficient; however in the

next remark we shall present a particular case in which this occurs.

Remark 2. In [2] Beals studied closed densely defined operators such that

R(A; A) exists and is polynomially bounded when Re A > C IImAla, for some

c > 0 and 0 < a < I; he essentially proved that in this situation A generates a

C-regularized semigroup {S} where C = e -£(-A)b, for E > 0, a ::; b < 1. If 1 l~O

we put U t = StC-1, t:2: 0, then

128 I. Cioranescu

II U tX II ~ M(t)11 C-1X II, X E C(X), with M(t) = II St II, t ~ o.

We can further adapt Sanekata's proof in [8] to obtain that A satisfies the property

ii) of the Main Theorem. For details and a more general case we refer to [1].

References

1. I. Cioranescu, Sur Ie probIeme de Cauchy au sens des ultradistributions, C. R. Acad. Sci. Paris, t. 300 Serle 1, No.7, 197-200, (1985).

2. R. Beals, Semigroups and abstract Gevrey spaces, J. Funct. Anal., 10, 300-308, (1972).

3. E. B. Davies and M. M. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc. 55, 181-208, (1987).

4. G. Da Prato, Semigruppi regolarizzabili, Ricerche Mat. 15, 223-248, (1966).

5. R. de Laubenfels, C-Semigroups and the Cauchy Problem, J. Funct. Anal., to appear.

6. I. Miyadera, On the generators of exponentially bounded C-semigroups, Proc. Japan Acad., 62, Ser. A, 239-242, (1986).

7. S. Oharu, Semigroups of linear operators in a Banach space, Publ. RIMS, Kyoto Univ., 7, (2),205-260, (1971).

8. N. Sanekata, Some remarks on the ACP, Publ, RIMS, Kyoto Univ. 11,51-65, (1975).

9. N. Tanaka, On the exponentially bounded C-semigroups, Tokyo J. Math., Vol. 10, No.1, 107-117, (1987).

University of Puerto Rico Department of Mathematics Faculty of Natural Sciences Box 23355 Rio Piedras, Puerto Rico 00931

Smoothing properties of heat semigroups

in infinite dimensions

Giuseppe Da Prato l

1. Introduction

Let us consider first a finite-dimensional problem

{

In 82u(t, x) Ut(t, x) = - LA;. 8 2 t ~ °

2 ;'=1 X;.

U(O, x) = <p(X), x = (Xl, ... , Xn)

(1)

where <p E Cb(Rn), the space of all uniformly continuous and bounded mappings Rn _ Rand '\1, ... ) An are positive numbers.

As well known, there exists a unique classical solution to (1), that can be represented by

(2)

where n

Sn(t)<p = IT Tk(t)<p, <p E Cb(Rn ) (3) ;'=1

and

(4)

Moreover Sn(t) is an analytic semigroup on Cb(Rn). We are interested to the infinite-dimensional generalization of (1); more precisely we are given a separable Hilbert space H with a complete orthonormal system {ed and a

1 Partially supported by the Italian National Project ~[URST "Problemi nonlineari nell'Analisi ... "

129

130 G. DaPrato

sequence of nonnegative numbers Pd. Then we set xk =< x, ek > and we consider the problem

{

I 00 82u(t, x) Ut(t,x) = -2 L Ak 8 2 , t? 0,

k=l Xk

u(O,x) = 'P(x), 'P E Cb(H),

(5)

where 8~k means the directional derivative in the direction of ek and Cb(H) is a suitable closed subspace of Cb(H), see §2 below. Obviously, existence for problem (5) is related to the existence of the limit

S(t)'P =: lim Sn(t)'P, 'P E Cb(H). n-+oo

(6)

As remarked in [2], a necessary condition for the limit in (6) exist for all 'P E Cb(H), is the following

(7)

This fact follows easily by computing Sn(t)'P when 'P(x) = e-tlx l2

From now on \ve assume that (7) holds, and we introduce the positive self-adjoint nuclear operator Q by setting

00

Q = L Akek 0 ek· k=l

Here we have used the notation

(xI8lY)Z = x < Y,Z >, x,y,z E H.

Then we write problem (5) as

{ Ut(t,x) = ~Tr [QUxx(t, x)]

u(O, x) = 'P(x), 'P E Cb(H),

(8)

Smoothing Properties 131

where Tr denotes the trace and the subscript x denotes the Frechet derivative with respect to x. In §2 we show, following [2], that if (7) holds then the limit in (6) exists for all y E Cb(H).

In §3 we consider the time-dependent problem

{ Ut(t,x) = ~ Tr [Q(t)uxx(t, x)], t E ]O,T]

u(O,x) = y(x), y E Cb(H),

(9)

where Q( t) are positive self-adjoint nuclear nuclear operators for all t E]O, TJ, and Q(-) is continuous on ]0, T] = {s E R : ° < s ~ T}. If

loT Tr [Q(t)]dt < +00, (10)

we prove that there exists an evolution operator U (t, s) so that the solution of (9) is given by u(t,·) = U(t, O)y.

As shown in [2], problem (9) naturally arises when one studies

{ Zt(t,x) = ~Tr [Qzxx(t,x)]+ < Bx,zx(t,x) >,

z(O,x) = y(x), y E Cb(H),

(11 )

where B : D(B) c H ~ H is the infinitesimal generator of a strongly continuous semi group etS in Hand Q is a self-adjoint positive operator, not necessarily nuclear. In fact, setting u(t, x) = z(t, etBx) and Q(t) = etBQetB', problem (11) reduces formally to problem (9), which can be solved, under assumption (10). In this case we can associate to problem (11) a semigroup Gs(t) defined by

(GB(t)y)(x) = (U(t, O)y)(etBx), t 2 0, x E H, y E Cb(H). (12)

§4 is devoted to prove a smoothing result; namely Gs(t)y E C!(H) for any y E Cb(H). This result was obtained in [4] by using probabilistic tools as the Cameron-Martin formula. Here we give a direct simple proof not involving probability. We remark that, when B is dissipative, a non probabilistic completely different proof was given in [2].

132 G. DaPrato

We remark finally that a probabilistic approach to problems (8) and (11), (and also to more general ones) is known from several years, see [3], [6], [7], [9], whereas a purely analitic approach is quite recent, see [1].

2. The heat semigroup

We are given a separable Hilbert space H, we denote by Ct(H) the set of all self-adjoint nonnegative nuclear operators on H and by Cb'(H) the set of all mappings from H into R that are Frechet differentiable togheter with their derivatives of any order, all derivatives being uniformly continuous and bounded. For any mapping c.p : H ~ R we set

11c.pllo = sup 1'P(x)l· xEH

Moreover, if 'P is k times Frechet differentiable we set

and k

Iic.pllk = 11c.pllo + L Ic.plh' h=l

-- --k We shall denote by Cb(H) (resp. Cb(H) ) the closure of Cb'(H) with respect to the norm 11·110, (resp. II· lid We recall that the space Cb(H), introduced in [2], is not dense in Cb(H), see [8], but it contains several functions of interest for the applications.

For any linear operator Q E .c.t (H) we denote by {e~} a complete or

thonormal system in H and by {)..~} a sequence of nonnegative numbers such that

Qe~ = )..~e~, k = 1,2, ...

Then, for any n = 1,2, ... we consider the analytic semigroups on Cb(H) defined as

(T:7(t)c.p)(x) = (il')..~t)-1/2 [:00 e_«J:·e?{.-e)2 c.p (x + (~_ < x,e~ »e~) d~ (13)


and n

S~(t)r.p = II T~(t)r.p, r.p E Cb(H). (14) k=l

We remark that the infinitesimal generator of the semigroup T;:O, n = 1,2, ... , is the operator (D~)2 where D~ represents the directional derivative on the direction e~.

The main result of this section is the theorem below whose proof may be found in [2]; however we sketch it for the reader's convenience.

Theorem 1 Let Q E £t(H); then the following statements hold

(i) For all r.p E C b (H), there exists the limit

SQ(t)r.p =: lim S~(t)r.p, in Cb(H). n-oo

(15)

(ii) SQ(t) is a strongly continuous semigroup of contractions in Cb(H) and its infinitesimal generator AQ is the closure of the linear operator A~ defined as

( 16)

(iii) For all positive integer k, C~(H) is an invariant subspace for the semigroup SQ(-), the restriction of SQ(-) to C~(H) is a strongly continuous semigroup of contractions in C~ (H) and its infinitesimal generator is the part of AQ in C~(H).

(iv) Let r.p E C~(H) and set u(t, x) = (SQ(t)r.p)(x), x E H, t 2: 0. Then u is continuous on [0, +oo[xH, moreover u(·, x) E Cl([O, +00[: R) for all

-2 x E H, u(t,·) E Cb(H) for all t 2: ° and

1 Ut(t, x) = "2 Tr [Quxx(t, x)], t 2: 0, x E H. (17)

Proof - In the proof we omit all superscripts Q for simplicity. If r.p E -2 Cb(H) we have

n

II Tk ( t) (Tn+l (t)r.p - r.p) k=l

134 G. DaPrato

It follows

and so 00 1 L: IISn+l(t)<p - Sn(t)<Pllo ::; 2t Tr QII<plb· n=l

Conclusions (i) and (ii) now follow from the fact that C~(H) is dense in Cb(H) ,the estimate

and standard arguments, see [2] for details. As far as (iii) is concerned, one has just to repeat the previous argument

- -k -2 -k+2 by replacing Cb(H) with Cb(H) and Cb(H) with Cb (H).

-2 We prove finally (iv). If <p E Cb(H) we have

By (iii) it follows that S(·)<p is differentiable in [O,+oo[ and equation (17) holds .•

We end this section by giving some results, which will be useful in §3, connecting different semigroups SQ (.) and SR(.), for Q, R E ..ct (H).

Proposition 2 Let Q, R E ..ct(H); then the following statements hold

(i) For all t, s ~ 0,

SQ(t)SR(S) = SR(s)SQ(t).

(ii) If <p E C~ (H), then

SQ(t)<p - SR(t)<p = lot SQ(t - s)SR(s)(AQ<p - AR<p)ds. (18)

Smoothing Properties

Proof - By recalling (13) we have

T~(t)Tf(s) = Tf(s)T~(t),

·which clearly implies (i). To prove (ii) set

u(t) = SQ(t)y, v(t) = SR(t)y, z(t) = u(t) - v(t), t:2: O.

Then z is the solution to the initial value problem

{ z'(t) = AQu(t) - [AQv(t) - ARv(t)]

z(O) = y,

and the conclusion follows from the variation of constants formula .•

135

Proposition 3 Let {Qn} be an increasing sequence in £i(H) strongly convergent to an element Q E £t(H). Then for any rp E Cb(H) we have

Proof- Since IISQn(t)yllo:S Ilrpllo, V rp E Cb(H),

it suffices to prove (19) for rp E C~(H). In this case we have

IIAQrp - AQnYllo = ~IITr [(Q - Qn)yxxllo

< ~Tr (Q - Qn)llyl12

1 ~ Q Q = 211rpl12 ~ < (Q - Qn)ek , ek >-+ 0

k=l

as n -+ 00. Now by Proposition 2 we have

which implies (19) •

(19)

136

Proposition 4 Let Q, R E £+(H); then we have

Proof- Since

it follows

Now, since

n

lim L[A~e~ 0 e~ + Afef 0 ef]x = (Q + R)x, V x E H, n-+<Xl

k=l

the conclusion follows from Proposition 3. •

3. Non autonomous problems

We are here concerned with the problem

{ Ut(t,x) = ~ T~[Q(t)uxx(t,x)], t E [O,T],

u(O,') = 'P E Cb(H),

under the following hypotheses

(i) Q(t) is nuclear Vt EjO,Tj.

(ii) Q(·)x is continuous in lO, Tl V x E H.

(iii) faT Tr [Q(t)]dt < +00.

We set

G. Da Prato

(20)

(21 )

(22)

L(t,s)x = it Q(a)xda,T ~ t ~ s ~ 0, x E H, (23)

clearly L(t,s) E £t(H), for T ~ t ~ s ~ O.

Theorem 5 Assume that (22) holds and set

U(t,s) = SL(t,s)(I), T ~ t ~ s ~ O. (24)

Then the following statements hold (i) For all T ~ t ~ s ~ a ~ 0 we have

U(t,s)U(s,a) = U(t,a), U(s,s) = I. (25)

-2 -2 (ii) U(t, s)<p E Cb(H) for all <p E Cb(H), T ~ t ~ s ~ O. (iii) U(·,s)<p E C1(ls,Tl;R) and

! [U(t,s)<pl = AQ(t)U(t,s)<p, Vt Els,Tl. (26)

Proof - (i) follows from Proposition 4, since

L(t,s) + L(s,a) = L(t,a)

for all T ~ t ~ s ~ a ~ O. (ii) follows from Theorem 1-(iii).

Let us prove (iii). Let <p E Ci(H) and t, t + h Els, TJ, then, recalling (18) we have

t(U(t + h,s)<p - U(t,s)<p) = k(SL(t+h,s)(I)<p - SL(t,s)(I)1')

As h tends to 0 we have by Proposition 4,

aU(t,s) r1 at l' = Jo SL(t.s)(1_ p)SL(t,s)(p) Tr [Q(t)1'xxldp.

and (26) follows. •

l38 G. Da Prato

4. Smoothing

4.1. Smoothing for equation (8)

Let Q E £t(H) and let SQ(t) be the semi group defined by (15). It is useful to introduce, for any positive integer k, the semigroup

00

S~)(t) = II T~ (t). (27) n=l,n;i:k

We remark that the existence of the infinite product in (27) follows from Theorem 1, applied to the Hilbert space HI generated by all element of the basis {e~} with the exception of ek.

Theorem 6 For all tp E C!(H), SQ(t)tp belongs toC!(H), and the following estimate holds

(28)

Proof - Let tp E C~(H). We first remark that SQ(t)tp E C!(H) by Theorem 1. Now setting Xk =< x, e~ >, k = 1,2, .. " we have

Since, as easily checked,

it follows

Ila~k (SQ(t)tp)lIo ::; (A~trI/21Itp"o, which is equivalent to

and now (28) follows immediately. •


Remark 7 By an easy density argument it follows, by Theorem 6, that SQ(t)c.p possesses all directional derivatives, with respect to e~, k = 1,2, .. that, in addition, belong to Cb(H). Similar results can be obtained from higher order directional derivatives. •

4.2. Smoothing for equation (11)

\Ve are given a self-adjoint nonnegative operator Q, not necessarily nuclear, and a closed linear operator B, infinitesimal generator of a strongly continuous semigroup etB . Then we set

Q(t) = etBQetB ·, L(t)x = L(t, O)x = l Q(s)xds, x E H. (29)

In this section we assume that Q( t) fulfills (22) and study the semigroup G B

defined by (12), which can be written, recall Proposition 4, as

Remark that, if Q is nuclear, then (22) is automatically fulfilled. We assume moreover

(31 )

As well known, hypothesis (31) is equivalent to the null controllability of the dynamical system

((t) = Bt(t) + Ql/2u(t), t ~ 0, ~(O) = x E H (32)

in any time t > 0, where ~(t) is the state and u(t) the control at time t, and it is always fulfilled if Q = I. In fact in this case, setting u(s) = -tesB, s E [0, tJ, one has ~(t) = O. For results on controllability see for instance [5J.

If (31) holds we can define a linear bounded operator

(33)

where (L(t)t 1/ 2 denotes the pseudo-inverse of (L(t))1/2.

140 G. DaPrato

Theorem 8 Assume that (22) and (31) hold. Then, for any rp E Cb(H) and -1

any t > 0, G8 (t)rp belongs to Cb(H) and

(34)

Proof--1

Let first rp E Cb(H). Then we have

(G8 (t)cp)x = et8·(SL(t)rp)x(etBx)

= r*(t)L1/2(t)(SL(t)rp)x(etBx).

By Theorem 6 it follows

so (34) is proved for rp E C!(H); the general case follows from the density of -1 -Cb(H) in Cb(H) .•

Remark 9 If Q = I the following estimate can be proved, see [4],

IIr(t)1I ~ r 1/ 2 sup IletBIi. sE[O,tJ

(35)


References

1. P. Cannarsa. G. Da Prato. A semigroup approach to Kolmogoroff equations in Hilbert spaces, AppJ. Math. Lett. 4,49-52 (1991).

2. P. Cannarsa. G. Da Prato. On a functional analysis approach to parabolic equations in infinite dimensions, preprint Scuola Normale Superiore di Pisa No. 139. (1992).

3. Yu. Daleckii. S. Fomin. MEASURES AND DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONAL SPACES, Kluwer (1991).

4. G. Da Prato. J. Zabczyk. Smoothing properties of transition semigroups in Hilbert spaces, Stochastics and Stochastic Reports., 35, 63-77 (1991).

5. G. Da Prato. J. Zabczyk. STOCHASTIC EQUATIONS IN INFINITE DIMENSIONS, Encyclopedia of Mathematics and its Applications, Cambridge University Press (1992).

6. B. Gaveau. Noyau de probabiliti de transition de certains opb:ateurs d'Ornstein Uhlenbeck dans les espaces de Hilbert C. R. Acad. Sc. Paris 293, 469-472 (1981).

7. L. Gross. Potential Theory in Hilbert spaces J. Funct. Analysis 1 , 123-189 (1965).

8. A. S. Nemirovski. S. M. Semenov. The polynomial approximation of functions in Hilbert spaces, Mat. Sh. (N.S), 92, 134,257-281 (1973).

9. M. A. Piech. A fundamental solution of the parabolic equation in Hilbert space J. Funct. Analysis 3 ,85- 114 (1969).

Scuola Normale Superiore Piazza dei Cavalieri 6 56126, Pisa, Italy

Locally Stable Dynamics for Reaction Diffusion Systems

W.E. Fitzgibbon* S.L Hollis J .J. Morgan *

1. Introduction

We shall be concerned with the dynamics of solutions to sys

tems of reaction diffusion equations. To be more precise we con

sider a weakly coupled semilinear parabolic system of the form:

au/at = Dt::.u + f(u) x E n, t> 0

au/an = 0 x E an, t > 0

u(x, 0) = uo(x) x E n.

(1.la)

(1.lb)

(1.lc)

Here the dependent variable u = (u 1 , •.• ~ U m) T is an m-dimen

sional vector and D is a diagonal matrix with strictly positive en

tries d;, i = I to m, along the diagonal. vVith a certain abuse of

notation we let t::. denote the vector Laplace operator and a/an represents a vector-valued Neumann boundary operator. We hope

that it will not introduce confusion when we use the same sym

bols subsequently in the text for scalar equations. The nonlinear

expression f( ) = (fl ( ) ... fmC )f shall be called the reaction

vector field. We shall require that

h( ) E C1 (Rm) for i = I to m.

143

G. R. Goldstein and J. A. Goldstein (eds.). Semigroups of Linear and Nonlinear Operations and Applications. 143-157. © 1993 Kluwer Academic Publishers.

(1.2)

144 W. E. Fitzgibbon et aI.

The initial data is required to be continuous; i.e.

(1.3)

Finally we stipulate that n is of class Ck and that an lies locally

on one side of n. Unless stated explicitly otherwise we shall as

sume the hypotheses delineated above hold through the course of

this text.

Reaction diffusion systems arise in a variety of scientific and

engineering contexts. Areas of application include chemical pro

cessing, population biology, semiconductor theory, contaminant

transport, geological modelling, reservoir flow, combustion the

ory and oncology. The qualitative behavior of such systems can

be difficult to understand but such an understanding can be cru

cial to a variety of scientific problems. We shall be concerned with

the standard engineering problem of describing the flow of system

(1.la-c) near equilibrium. Equilibrium solutions of (1.la-b) are

known to satisfy weakly coupled semilinear elliptic systems of the

form

-Db.w = few) x En awjaT] = 0 x E an.

(1.4a)

(lAb)

Spatially homogeneous solutions to (1.4a-b) are provided by zeros,

or critical points, of the vector field. Thus if Zo E Rm is such that

f(zo) = 0 E Rm

then Zo is a solution to (lAa-b). However, cf. [13], it is well

known that spatially inhomogeneous solutions may also exist.

We shall follow the following organization for the remainder

of this paper. The next section overviews the familiar notion of

Locally Stable Dynamics 145

invariant rectangles for reaction diffusion equations and demon

strates the technique of truncating t.he system in the vicinity of

stationary solutions. The third section introduces a subsidiary

notion of stability of steady-state solutions and outlines a proce

dure which demonstrates that this subsidiary notion of stability

in fact implies stability. In the fourth section we adapt Lyapunov

methods to examine the stability of constant steady-states and

we conclude with a section of examples.

2. Invariant Regions and Truncated Systems

The material overviewed in this section is well known in the

literature and the reader is referred to [14] for an excellent and de

tailed exposition. A closed subset M is said to be a forward invari

ant region for (lola-c) if uo(x) EM for x E n implies u(x, t) E M

for x E n, t > O. If

M = II x··· x Im (2.1)

where each Ii is a closed (possibly unbounded) interval and the

vector field f( ) does not point out of M along 8M then M is

an invariant region for (lola-c). In most applications we are con

cerned with nonnegative quantities and therefore M will usually

denote the positive orthant, R+.

Throughout the remainder of this paper we assume M is a

forward invariant region for (lola-c). Suppose w is a solution

to (lo4a-b) such that w(x) E M for all x E n. Let B be an

m-dimensional cube such that w( x) E int B for all x E n, and

let 1] = inf dist( w( x), B). If 1] > 0 rename B as b1 (1], w) and xEQ

let bz(7],w) be the m-cube concentric to b1(1],w) with twice the

diameter. We set Ml = M n b1 (7], w) and Mz = M n bz( 7], w).

146 W. E. Fitzgibbon et al.

If we mollify the characteristic function of bl (7], w) we can

produce a cutoff function <P.",w E coo(Rm, [0, 1]) such that

<P.",w(v) = 1 for v E bl (7],w)

<P.",w(v)=O for vERm-~(7],w).

(2.2)

We truncate the vector field by componentwise multiplication by

<P.",w; i.e. we define 1[7], w] = (f;[7], w])~1 by

fi [7], w]( u) = <P.",w( u )fi( u).

We may now consider the following truncation of (l.la-c):

fJv/fJt = D~v + J[7], w](v) x E n,t > 0

fJv/fJn = 0 x E fJn, t > 0

We have the following existence and containment result:

(2.3)

(2.4a)

(2.4b)

(2.4c)

Lemma 2.5. System (2.4a-c) has an unique classical solution on

IT x [0,(0). Moreover;

(i) v( ,t) E M2 for t ;::: 0, and

(ii) if v( ,t) E 1111 for 0 ::; t < T til en

vex, t) = u(x, t) for x E IT, 0::; t < T,

where u( ) is the solution to (l.la-c).

Proof. We observe that 1112 is a bounded invariant region for

(2.4a-c) because the vector field 1[7], w] is identically zero exte

rior to b2[17, w], and hence does not point out of }.f. Therefore

solutions t.o (2.4a-c) exist globally and remain confined to 1112 for

all time. Classical parabolic uniqueness theory together with the

observation that f[ 7], w J = f on b1 ( 7], w) guarantee the second

assertion. •

3. oo-r Stability

We introduce a subsidiary notion of uniform stability and

demonstrate that this notion may be used to guarantee uniform

stability of steady-state solutions to (1.1a-c). Our analysis will

involve the standard Lp(n) spaces with p ~ 1. However, we shall

also want to consider Lp(n) spaces with /lu/lp,n = [f lulPdxj1/p

with 0 < p < 1. We point out that if 0 < p < 1, II/lp,n is not a

norm. However, it may be used to define a complete metric space.

\Ve now introduce our notation of uniform stability:

Definition 3.1. Let Zo E M be a critical point for the vector

field f = (fi)~l' Then Zo is said to be stable with respect to M

iffor all $ > 0 there exists a b > 0 so that uo( x) E M for all x E n and /luo; - Zo; 1100,n < 8 for i = 1 to m imply

(i) A classical solution to (l.la-c) exists on n X [0,00), and

(ii) /lUi(', t) - Zo; lloo,n < t: for i = 1 to m and t > O.

A critical point Zo E M which is stable with respect to M is said

to be asymptotically stable with respect to M if there exists a

b > 0 so that ifuo(x) E M for all x E n with /luo; - ZOi/loo,n < 8

for i = 1 to m, then we have

(iii) lim IIUi(', t) - zo·lIoo n = 0 for i = 1 to m. t~oo ' ,

Our motivation for defining stability with respect to M is

that we shall wish to consider critical points belonging to aM. We now introduce our notion of 00 - r stability.

148 w. E. Fitzgibbon et al.

Definition 3.2. Let Zo E M be a critical point of the vector field

f = (fi)~l and let r > O. Then Zo is said to be oo-r stable with

respect to M if for all c > 0 there exists a 0 > 0 so that Uo E M

and Iluo; - ZO; 1100,n < 0 for i = 1 to m imply

(i) a classical solution to (l.la-c) exists on 51 x [0,00), and

(ii) Ilui(·, t) - zo.llo,oo < c for i = 1 to m and t > O.

A critical point which is 00 - r stable with respect to M is said

to be asymptotically oo-r stable with respect to M if there exists

a 0> 0 so that ifuo(x) E M for all x E 51 with Ilui - ZOj 1100,n < 0

for i = 1 to m, then we have

(iii) tlim Ilui(·, t) - ZOj IIr,co = 0 for i = 1 to m . ...... 00

We are now in a position to provide the following theorem

which is the foundation of our development.

Theorem 3.3. Let w be a solution to (l.4a-b) such that w(x) E

M for all x E 51. If r > 0 and w( ) is a oo-r stable steady-state

solution of (2.4a-c) with respect to M then w is a uniformly stable

solution of (1. 1 a.-c) with respect to M.

Discussion of Proof. This theorem requires a lengthy boot

strapping argument and the reader is referred to [2] for details.

First one uses the boundedness of solutions to (2.4a-c) to argue

that oo-r stability implies 00-2 stability. The difficult part of the

argument involves the bootstrapping of 00-2 stability to uniform

stability. Here one uses energy arguments in conjunction with

the parabolic regularity estimates of Ladyzhenskaja, Solonnikov

and Ural'ceva [7] and a fractional Sobolev embedding theorem

appearing in Amann [1]. •


If one has a priori knowledge of the boundedness of solutions

to (lola-c), then one may dispense with the truncated systems

(2.4a-c) and directly conclude that OO-T stability of steady-state

solutions to (lola-c) implies uniform stability. Finally, we point

out that only the boundedness of the steady-state is essential and

the arguments of Theorem 3.3 can also be applied to determine

the stability of other distinguished bounded solutions such as pe

riodic solutions and almost periodic solutions.

4. Diffusively Convex Lyapunov Functionals

As an application of the foregoing theory we discuss the Lya

punov stability of critical points or zeros of the vector field f( ). It

should be clear that the system of ordinary differential equations,

du/dt = feu)

u(O) = Uo

(4.la)

(4.lb)

determines spatially homogeneous solutions to (lola-c). Critical

points of f give steady-state solutions to (4.la-c) and spatially

homogeneous steady-state solutions to (lola-c).

The most common tool for analyzing the local stability of

critical points for systems of ordinary differential equations of the

form (1. a-b) is the principle of linearized stability. If all the

eigenvalues of the derivative of f at '::0 have negative real part

then Zo is locally asymptotically stable. If on the other hand,

some of the eigenvalues have negative real part and some have

positive real part the critical point Zo is unstable. These ideas

carryover to the context of semi linear parabolic equations; see,

e.g., [5]. In the case of nonhyperbolic critical points, however,

linearization methods do not apply.

150 W. E. Fitzgibbon et aI.

Questions of nonlinear sta.bility are frequently resolved by

Lya.punov's direct method. Roughly speaking a Lyapunov func

tion V is a nonnegative functional which is defined and continu

ously differentiable in a neighborhood of a critical point Zo and is

uniquely minimized in that neighborhood by Zo. If

V(u) = 8V(u)f(u) ~ 0 (4.2)

in this neighborhood, then it follows that Zo is a stable critical

point. Asymptotic stability can be deduced from conditions such

as

i'(u) < -aV(u) (4.3)

for some a > O. In certain cases the existence of Lyapunov func

tionals satisfying (4.2) in a neighborhood of a critical points of a

system of ordinary differential equations carries over to the con

text of the associated reaction-diffusion system. For this reason,

we introduce the notion of D-diffusively convex Lyapunov func

tionals for reaction-diffusion systems.

Definition 4.4. Let D be the matrix of diffusion coefficients for

(l.la) and suppose that M is a forward invariant set for (l.la-c).

If Zo E M is a critical point of f we say that a nonnegative func

tional V is a D-diffusively convex Lyapunov functional around Zo

provided that the following conditions hold:

(i) There exists a ~ > 0 so that V E C2(M n B{(zo), R+).

(ii) There exist constants r > 0 and K > 0 so that V ( u) ~ m

K 2:: lUi - zoX for u E Be(zo) n M. i=l

(iii) V(zo) = O.

(iv) The matrix D82V(u) is nonnegative definite for

u E Be(zo) n M. (Here 82V(u) is the Hessian matrix


of V).

(v) aV(u)f(u) s 0 for u E B{(zo) n M.

We remark that conditions (i), (iii) and (v) are essentially those

which define a Lyapunov functional for (1.1a-b) around Zo and

that conditions (ii) and (iv) represent an additional strengthening

of the concept. If the functional V is separable; i.e.,

n

F(u) = LVi(Ui), (4.6) i=l

then we may insure (iv) by assuming that Vi" (ud ~ o. In general,

however, convexity of V does not suffice for condition (iv). It is

relatively straightforward to see that D-diffusively convex Lya

punov functionals guarantee the persistence of stability of critical

points. We have the following theorem:

Theorem 4.6. Let Zo E M be a critical point for the vector neld

f and M be a forward invariant set for the semilinear parabolic

system (l.la-c)' If there exists a local D-diffusively convex Lya

punov function V for f around Zo, then Zo is a stable steady-state

for (l.la-c) with respect to M. ]v[oreover, if V also satisnes (4.3),

then Zo is asymptotically stable with respect to 111.

Proof. We choose ry > 0 so that the cube C21j(zo) ~ B{(zo) and

construct the truncated vector field j[ry, zo] as in (2.2), (2.3). If

vo(x) E C21j (zo) n M for x E n it is immediately verified that

av(v(x, t))f(v(x, t)) = aV(v(x, t))f[ry, zo](v(x, t)) SO. (4.7)

If we multiply the ith component of (2.12a) by aV(V)/aVi we

obtain

(av( v)/ avdavi/ at = die av(v)/ avil6vi+( ave v )/aVi).fi[ry, zo]( v).


If we integrate this expression on the space-time cylinder and sum

the components, we observe that

in V(v(x, t))dx = -iT in ('\lv)T D82V(v)Vvdxdt

+ it in 8V(v)J[7J,zoJ(v)dx + in V(vo(x))dx.

Hence by virtue of conditions (iv) and (v) in Definition 4.4 we

have

in V(v(x, t))dx ::; in V(vo(x))dx. (4.8)

Using (4.8) and the coerciveness of V we get

K [~IIV,(-' I) - zo.,II •. o 1 $ [1 V(v(x, l)dXr (4.9)

~ ::; [In 1'(vo(x,t))dx] r

$ p (~ Ilvo, - Zo, 1100.0 )

for some continuous p with p(O) = 0 and p(s) > 0 for s > O. This

will insure oo-r stability with respect to M, and from Theorem 3.3

we may conclude that zo is stable with respect to M. Finally, in

case (4.3) holds, one has

in V(v(x, t))dx ::; e-at in V(vo(x))dx, (4.10)

and from this follows the asymptotic stability assertion. •

In view of Theorem 4.6 one can be naturally lead to the

attempt of using D-diffusively convex Lyapunov functions to an

alyze the stability of spatially non-homogeneous steady-state so

lutions. The following simple proposition squashes this endeavor

for large classes of dynamical systems.

Proposition 4.11. Let M be a forward invariant set for (l.la-m

c) and let V ( v) = "L Vi (vd be a nonnegative separable function i=1

which satisfies the defining hypotheses of Definition 4.4 except

possibly (ii), for all points of M. If W = (WI, ... , Wm f E M is a

solution to (l.4a-b) then the following are true:

(i) If there exists Q > 0 so that V/, (vd > Q for all V

(vl, ... ,vm f E l'vl, then f(w) = o. m

(ii) If 1/( v) = I: CiVj, cfV( v )f(v) :s: 0 and M <;;; Wi.', then j=1

m

there exists a k 2 0 so that y(x) = I: CjdjWi(X) = k for i=1

all x E n; i.e., w(x) belongs to a closed bounded subset

of the hyperplane {v I r;cjdjvj = k} n R+'. (4.11 )

Proof. In the first case we multiply the ith component of (2.35a)

by V/ ( Wi) to obtain

-dV'(w·).6.W· = V'(w)f·(w) 1 1 1 1 1 1 1 • (4.12)

If we sum these terms and integrate on n we have

Consequently,

(4.14)

and we may conclude that each Wi is a constant. Therefore, be

cause W = (WI .. ' W" f is a solution to (1.4a-b), we must have

fi( w) = O. If we follow the same train of reasoning for the second

case we may observe that -.6.(~cidiW;) :s: o. The fact that AI

is required to lie in R+, implies that ~diWi 2 0, and hence we

conclude from maximum principles that \7(~cidiWi) vanishes and


~cidiWi( x) = k for some constant k ~ O. Thus w( x) lies in the

hyperplane {v I ~cidiVi = k}. Continuity of w implies that its

range is closed and bounded. •

We conclude this section with a discussion of an example aris

ing from the theory of chemical reactions. Differential equations

which describe the dispersion and reaction of m-chemical species

are generally of the form

au/at = Dt::.u + f(u) (4.15)

where the ith component of the dependent variable u

(U1' ... ,U m f represents the concentration density of the i th chem

ical species. The vector field f = (I; )~1 is assumed to be in each

component a polynomial function of the components of u and is

intended to model the chemical reaction kinetics. Groger, in his

study of dissipative chemical reactions, [4], introduced the follow

ing hypothesis.

(G) There exists a vector e = (e1, ... , em f with each ei > 0 so

that f(e) = 0 and

m

L.f;(u)log(u;/ei) ~ O. (4.16) i=1

m

The quantity L fi(u)log(u;/e;) is known to have the physical ;=1

interpretation of being a suitably scaled rate of chemical dissipa-

tion, and work on the mathematical theory of reaction networks,

[7], confirms that many nontrivial systems satisfy this hypothe

sis. If the chemical species are required to remain confined to a

reaction vessel for all time the appropriate boundary conditions

are given by

au/an = 0 ( J:, t) E an x (0, 00 ). ( 4.17)


Finally, a condition of the form

fi( u) 2 0 for u E R+, with Uj = 0 (4.18)

together with the maximum principle insures that R+ is a forward

invariant set for (4.15). We have the following proposition:

Proposition 4.19. We consider (4.15) together with the bound

ary conditions (4-17). If all the conditions describing a dissipative

chemical reaction outlined above hold, then the steady-state U = e

is uniformly stable with respect to R+. Moreover, the elliptic sys

tem

-D.0.w = f(w) x E n

8wj8n=O xE8n

bas no spatially inhomogeneous positive solutions.

Proof. vVe define

m. m

l1(u) = L V;(u.) = L(uilog(ujei) -llj + ei) i=l i=l

(4.19a)

(4.20b)

(4.20)

and verify that all conditions of Definition 4.4 hold locally about

e. Consequently, Theorem 3.4 implies that e is uniformly stable.

An argument analogous t.o the one of Proposition 4.12 insures the

nonexistence of positiye spatially inhomogeneous steady-states .

• We remark that [2J gives a similar analysis for the stability

of constant steady-states of the general diffusive Lotka-Volterra

system outlined in [9J. The reader will find additional use of the

structure of D-diffusively convex Lyapunov functions in [3], [10J

[11], [12J.


References

1. H. Amann, Existence and regularity for semilinear parabolic

evolution equations, Annali Sco'U.la Nosmale Superiore - Pisa,

Serie IV, Vol. IX 593-676 (1984).

2. W. Fitzgibbon, S.1. Hollis and J.J. Morgan, Stability and

Lyapunov functions for reaction-diffusion systems, Preprint.

3. \V. Fitzgibbon, J. Morgan and R. Sanders, Global existence

and boundedness for a class of inhomogeneous parabolic equa

tions, J. Nonlinear Analysis, to appear.

4. K. GrageI', On the existence of steady-states of certain reac

tion diffusion systems, Arch. Rat. Mech. Anal., 1: 297-306

(1986).

5. D. Henry, Geometric Theory of Semilinear Parabolic Equa

tions, Lecture Notes in Mathematics, 840, Springer-Verlag,

Berlin, (1981).

6. S. Hollis. R. Martin and M. Pierre, Global existence and

boundedness in reaction-diffusion systems, SIAM J. Math.

Anal., 18: 744-761 (1987).

7. F. Horn and R. Jackson, General mass action kinetics, Arch.

Rat. Mech. and Anal., 47: 81-11G (1972).

8. O. Ladyzhenskaja, V. Solonnikov and N. Ural'ceva, Linear

and Quasilinear Equations of Parabolic Type, AMS Trans.,

Vol 23, Amer. Math. Soc., Providence, R.I., (1968).

9. A. Leung, Systems of Nonlinear Partial Differential Equa

tions, Kluwer Academic Publ., Boston, (1989).

10. J. Morgan, Boundedness and decay results for reaction

diffusion systems. SIAM J. Math. Anal., 21: 1172-1181

(1990).


11. J. Morgan, Global existence for reaction diffusion systems,

SIAM J. Math. Anal. 20: 1128-1144 (1989).

12. J. Morgan, Global Existence for a class of quasi linear reaction

diffusion systems, Preprint.

13. J. Murray, Mathematical Biology, Springer-Verlag, Berlin

(1989).

14. J. Smoller, Shock Waves a.nd Reaction Diffusion Equations,

Springer-Verlag, Berlin, (1984).

Department of Mathematics

University of Houston

Houston, Texas 77204-3476 USA


Armstrong State College

Savannah, Georgia 31419-1997, USA


Texas A&M University

College Station, Texas 77843-3368, USA

*These authors gratefully acknowledge the support of NSF

Grants DMS 9207064 and DMS 9208046 respectively.

Global Dynamics of Singularly Perturbed Hodgkin-Huxley Equations

W. Fitzgibbon 1

M. Parrott Y. You

In their well-known work, Hodgkin and Huxley considered the following model for nerve impulse transmission across an axon:

(1)

8m at = (moo - m) ITm (2)

8h at = (h oo - h) ITh (3)

(4)

Here x represents the longitudinal distance along the axon, t is time; V is the electrical potential in the nerve; m (V), h (V), n (V) are chemical concentrations of Na ) J( and other (leekage) ions which are nonnegative and are nonlinear functions of V; gNa) gK, gL are the maximum conductances of these ions) VNa ) VK , VL are the constant equilibrium potentials for these ions; moo, hoo, noo are the steady state values; and Tm, Th, Tn are relaxation times.

ISupported by NSF Grant No. DMS 9207064

159


160 W. Fitzgibbon et aI.

A more thorough examination of the derivation of equation (1) reveals that (1) is actually an approximation of the equation

(I')

where <: is a positive constant which represents inductance, which is assumed small, and thus the terms involving E are usually ignored. Some numerical studies done by Lieberstein [8] suggest that solutions of (I'), (2)-(4) (with appropriate initial and boundary conditions) do converge, as E -+ 0, to solutions of (1), (2)-(4). Thus, some justification is provided for ignoring inductance terms in the usual Hodgkin-Huxley system.

The mathematical justification of these numerical studies is provided in a recent paper of Fitzgibbon and Parrott [5]. Instead of (I'), (2)-(4), they consider the following simplified system with Neumann boundary conditions and the given initial data:

ow at = -kw + g Cu, w), t? 0, x E (0, 1)

au OU o = ox (0, t) = ox (1, t), t ? °

U (x, 0) = Uo (x), x E (0, 1)

[hI at (x, 0) = UI (x), x E (0, 1)

w (x, 0) = Wo (x), x E (0, 1)

Global Dynamics

Uo, UI, Wo E Coo (0, 1) ° = u~ (0) = u~ (1) = U~' (0) = U~' (1) ° = u~ (0) = u~ (1) = U~' (0) = U~' (1) ° = wb (0) = wb (1) = W~' (0) = W~' (1).

161

Here k is a positive constant, iI, 12, h are continuous, polynomial bounded functions, and 9 is a continuous and uniformly bounded function. (No restrictions on the signs of these functions are made in [5].)

Corresponding to (HH)" the usual Hodgkin-Huxley system takes the following form:

ow at = -kw + 9 (u, 11,'), t? 0, x E (0, 1)

ou ou ° = -0 (0, t) = -0 (1, t), t ? ° x x

u (x, 0) = Uo (x) , x E (0, 1)

10 {x, 0) = wo{x), x E (O, 1)

uo, Wo E Coo (0, 1) ° = u~ (0) = u~ (1) = U~' (0) = U~' (1) o = 11,'~ (0) = w~ (1) = W~' (0) = W~' (1) .

The following lemma gives the boundary behavior of w:

Lemma 1 (5, Lemma 2.8) . If (u (-, .), 10 (., .)) is a classical solution to (HH), or (HH), then

ow ow [AO = ox (0, t) = ox (1, t) , t ? 0.

162 W. Fitzgibbon et al.

Fundamental work on the traditional Hodgkin-Huxley equations has been done by Evans [1], [2], [3] and Evans and Shenk [4], and the existence of a unique classical solution of (HH) on 0 ~ t ~ T, and any T > 0, follows from this work. The existence of a unique classical solution of (HH)f for each € > 0 on 0 ~ t < 00 is shown by using semigroup theory in [5]. We state below the main result of [5], which establishes the convergence of (HH)f to (HH) as € --+ O. This result is obtained by using energy arguments, a priori Sobolev estimates, and recent abstract convergence results of Najman [10]. Here 11.11 00 denotes the supremum norm of C [0, 1].

Theorem 1 (5, Theorem 6.12) . Let (u(., .), w(.,.)) be the solution to (HH) and let (u f (', .), W f (', .)) be the family of solutions to (HH)f' Then

lim lIu( (', t) - u (., t)lIoo = 0 (-+0+

and

lim IIwf (., t) - W (., t)lIoo = 0, <-+0+

t E [0, T] , for any given T > O.

In the present work we seek to understand the long-term behavior and global dynamics of the perturbed Hodgkin-Huxley system. Specifically, we will show the existence of global attractors Af for the perturbed system for each € sufficiently small. Properties of these attractors (for example, the fractal dimension and Hausdorff dimension), and the relationship of Af to A, the global attractor corresponding to the unperturbed Hodgkin-Huxley system, will be the subjects of future work. We note that the existence of the global attractor A follows from known results of Smoller [11] and Temam [12].

We begin by rewriting slightly the equations in (HH)< to reflect more accurately the physical system modelled by equations (I'), (2)-(4). We then give physically realistic assumptions which will be sufficient to guarantee our results.

Global Dynamics 163

From now on, consider the singularly perturbed Hodgkin-Huxley equations:

(t, x) E R+ X (0, 1),

Wt = -hI (u)w+h2(u),

with the Neumann boundary conditions

Ux(t, 0) = ux(t, 1) = 0, t ~ 0,

and initial conditions

(SPHH)

(NBC)

u(O, x) = uo(x), Ut (0, x) = ut{x) , w(O, x) = wo(x) , x E [0, 1].

Assumptions: We assume that

1) N > 0 is a fixed constant, to > 0 is a constant which can be arbitrarily small.

2) it (w) ~ a with a > 0 a small constant. 3) hI (u) ~ b with b> 0 a small constant, and h2 (u) ~ O. 4) II, 12, and 13: R --t R and their derivatives are continuous and

polynomial bounded. 5) hI and h2: R --t R are uniformly bounded; h~ and h~ are locally

bounded. 6) w* ~ Wo (x) ~ 0 for x E [0, 1], where w* > 0 is a fixed constant.

We emphasize that these assumptions concerning the nonlinearity can be actually verified in the case of the real model system. As a consequence of these assumptions, we know that

w (t, x) ~ exp ( - j h, (u (u,x)) dU) Wo (x)


+ I h, (u ("x)) exp ( - ! h, (u (o,x)) dO) d, and therefore,

* G (h2) ( ) , 0::; w (t, x) ::; w + -b- (denoted by Gw ), for t, x E RT X (0, 1).

Hence, by the assumptions, there exists a uniform constant Gj > ° such that

sup {lfi (u.; (t, x))1 : (L x) E R+ X (0, I)} ::; Gj , i = 1, 2, 3.

From a minor adjustment to the work in [5], for any

Uo = CUD, UI, wo) E E = HI (0,1) X [2 (0,1) X [2 (0,1),

the strong solution U (t) (x) = (u (t , x), Ut (t, x), w (t, x)) of the abstract evolution equation

(5)

exists uniquely and globally for t E [0, (0), where G, : D (G,) = [Hh (0, 1) n HI (0, 1)] X HI (0, 1) X [2 (0, 1) -t E is the linear operator and F, is the nonlinear mapping, similar to (4.2) and (4.3) in [5].

Let us recall the definition of a global attractor for a semiflow (1 (t, y) : R + x Y -t Y in a Banach space Y. Let the semigroup associated with (1 be denoted by S (t) : Y -t Y. Namely, S (t) Y = (1 (t, y).

A set A C Y is called a global attractor for the semiflow (1 if it is compact, functional invariant (i.e. 5 (t) A = A, for any t ;::: 0), and dist (5 (t) E, A) -t 0 as t ---+ 0, for any bounded set E in Y.

As another related notion, a set BeY is called an absorbing set for the semifiowa if a flux of trajectories started from any given bounded set Z C Y enter into the set B after a transient time duration which is uniform in Z, and will stay in B forever.

We refer a Basic Theorem on the existence of a global attractor for a semiflow to [12] (p.23, Theorem 1.1). Below is the version of this theorem,

Global Dynamics 165

which we will apply to the semiflow generated by the nonlinear evolution equation (5) which is derived from the original (SPHH) equations.

Basic Theorem. Assume that the semigroup S (t) in a Banach space Y satisfies the following condition: S (t) = Sl (t) + S2 (t) for every t ;::: 0, where Sl (t) and S2 (t) : Y ...... Y satisfy (i) for every bounded set Z C Y, sup {liS (t) vii: v E Z} ...... 0 as t ...... +00, (ii) S2 (t) is uniformly compact for t large, i.e. for every bounded set Z C Y, there is to ~ 0, such that

U S2 (t) Z is precompact in Y. t~to

Then there exists a global attractor A in Y for this serniflow cr, provided that there exists a bounded absorbing set B. Moreover, A = w (B) where w ( .) is the w-limit set. 2

Remark. We note some facts about our system (SPHH) with the Neumann boundary conditions, which might be expected to present obstacles to showing the existence of absorbing sets. First, the (SPHH) system can be viewed as a partially dissipative system of a semilinear hyperbolic equation coupled with ordinary differential equations which have no diffusion term. While there have been s~me results showing the existence of attractors for partially dissipative semilinear parabolic systems, cf. [9], there are no such results for hyperbolic-type systems as far as the authors are aware. Even for some significant infinite dimensional dynamical systems which are physically considered as dissipative (e.g. 3-dimensional N avier-Stokes equations), the existence of absorbing sets is unknown. Secondly, the Neumann boundary conditions associated with dissipative wave equations or damped elastic Petrovsky equations often lead to difficulties in bounding the mean value of the solutions in the region if there is no additional assumption, d. [12, Chapter IV].

Here we conduct a priori estimates as follows to show that there exist absorbing sets in the space E. Let 1.1 and (.,.) denote the norm and the inner-product in U (0, 1). Taking the inner-product of the first equation in (SPHH) with Ut and pu in U (0, 1) , we have


(6)

and

(7)

Sum up (6) and (7) to obtain

tit {t IUtl2 + luxl2 - 2tp (Ut, U) + N p Iun

{ 2 2 1 2

+ (N-tp)IUtl +plUxl +t[h(w)IUt(t,x)1 dx

+ } [Jd w) + tp f3 (w) 1 U (t, x) Ut (t, x) dx (8)

+ pI f,(w) I:(t, x)I' dx - p(J, (w), u) - (h (w), u,)} ~ O.

Define r (t) by

r (t) = t IUt!2 + luxl2 - 2f.p (Ut, u) + N P luI2 .

Then we see that

{ 2 2 1 2

(N-tp)IUtl +pluxl +t[f3(W)lut (t,x)1 dx 1

+ flfl (w) + tph(w))u(t, x)udt, x)dx (9)

+p 1 f, (w) lu ;t, u)I' dx - p (f, (w), u) - (j, (w), u,)} - Kf (t)

1

~ (N - tp - til:) IUtl 2 +(p - ... ) IUxl2 + J Udw) + tpf3 (w) + 2tp ... ] u (t,x) udt,x) dx o

Global Dynamics 167

Now we have two undetermined constants p > 0 and K, > 0 that can be arbitrarily chosen. Besides the intrinsic perturbation parameter f > 0 is a small fixed number. Our idea in making the subsequent ordered choice is to show the following absorbing result.

Theorem 2 There is a constant fO > OJ such that for the semiflow Ve generated by (SPHH) or the formulated abstract evolution equation (5) with parameter 0 < f ~ fOJ there exists an absorbing ball En in the space E.

Proof. In the last inequality of (9), let us first choose and fix p > 0 such that

pa _ N _ 4 ( C f + 1)2 > 0 2 N _.

Then choose and fix K. > 0 such that

(10)

(11)

which implies that p - Il. ;::::: 0, and 0 < pll. ~ 1. Thus, from (9) it follows that

1

+ J [JJ(w) + fph (w)Ju (t, x) udt, x) dx a

+p I t. (w) I" (t, x)I'dx - p (f, (w), u) - (J, (w), u,) } -.r (t)

;:::::_(Cf )2G+ ~), where 0 < f ~ fa, and fa > 0 is a constant such that

fa (p + ~) ~ ~ and fa max {p, I} ~ 1. (13)

Substituting (12) into (8), we get

(14)

where Il. > 0 is any constant satisfying (11) in which p > 0 satisfies (10), and the right-hand side is a constant denoted by

Therefore,

Global Dynamics 169

C1 r(t) ~ r(0)exp(-2~t) + 2~' for t ~ o. (15)

The regularity of the initial data allows pointwise evaluation and it is easy to see that r (0) is a functional of (uo, U1) and

r (0) ~ max {100, 1, N P + p2} lI(uo, udll~lXL2 . (16)

On the other hand we have

(17)

where, by (13),

N p - 2fp2 = 2p (~ - lOP) ~ 210.

Therefore,

10 2 2 2 10 2 r(t)~2Iutl +Iuxl +2flul ~211(u(t),ut(t))IIH1XL2. (18)

Combining (15) with (16) and (18), we obtain

This means that

C1 +-, for t ~ o. f~

lim sup II(u (t), udt))lI~lX£2 ~ C1 . t-+oo f~

(20)

Besides it is known that Ilw (t, .)1I~2 ~ (Cw )2. Incorporating this with (20), finally we can assert that the ball BR C E centered at the origin and with radius R,

(21 )

is an absorbing set of the semiflow V f with 0 < t ::; to. The proof is completed. 2

Theorem 3 . There exists a global attractor M C E for the semifiow V f

generated by (SPHH) or the formulated abstract evolution equation (5) with parameter 0 < t ::; to, where to > 0 is a uniform constant.

Proof. Since we have shown the existence of a bounded absorbing set by Theorem 2, it suffices to show that the solution semigroup Sf (t) associated with the semifiow V( satisfies the decomposition property required in the Basic Theorem.

For this purpose, we write the solution W (t) of the second equation of (SPHH) as w (t) = WI (t, .) + W2 (t, .) with

WI (t, x) = wo(x)exp (-1 hI (u(s, X))dS) ,

W2 (t, x) = I hz ('11 (s, x)) exp ( -1 hdu (a, x)) dO' ) ds, (22)

and write the solution '11 (t) of the first equation of (SPHH) as '11 (t) = VI (t, .)+ Vz (t, .) with VI (t, .) satisfying the following linear equation:

Wtt + (N + th (w)) Vt - Vxx + av = 0, with (NBC), VI (0) = '110, Vlt (0) = '111·

(23)

Definite component families of operators Sd (t) and Se2 (t) as a decomposition of Sf (t) by

Sel (t) ~ ('110, '111, wo) --t (VI (t, .), ~Vl (t, .), WI (t, ·)l ' (24) 5<2 (t). ('110, '111, wo) --t (V2 (t, .), 8iV2 (t,.), W2 (t,.) .

It is not difficult to show that 5el satisfies the condition (i) in the basic theorem. Here the details are omitted. The remaining task is to check that 5 f2 satisfies the uniform compactness for t large, the condition (ii) in the basic theorem.

The component V2 satisfies the following equation: (we denote V2 by v for simplicity)

Global Dynamics 171

Wtt + NVt + ff3 (W) Vt - Vxx + fdw)u - aVI - f2 (w) = 0, with (NBC), V (0) = 0, Vt (0) = 0,

(25) where VI is the linear component and u is the original solution of the first

equation. From the relation V (t) (= V2 (t)) = u (t) - VI (t) and the properties possessed by u and Vb there is a uniform bound C2 (r) > 0 such that

II(V(t), vdt))lI~lX£2 ~ C2(r), t? 0,

for any (uo, UI, wo) E Br C E. Taking the inner-product of (25) with -Vxxt in L2 (0, 1), we get

fd 2 2 1d 2 '2 dt IVxtl + N IVxtl + '2 dt Ivxxl (26)

d 1 lId 1

-f dt J h (W) VtVxx dX+f J h (W) VttVxx dX+f J f~ (W) WtVtVxxdx+ dt J aVI vxxdx o 0 0 0

dId 1 1

- dt J fl (w) uvxxdx + dt J i2 (w) vxxdx - a J Vltvxxdx o 0 0

1 1

+ J fl (w) Utvxxdx + J f{ (w) Wtuvx:cdx - J f~ (w) WtVxxdx o 0

d { 2 2 1 =di {f/2)IVxtl + (1/2) Ivxxl -f[f3(W)Vtvxxdx

+ j aVlvxxdx - j it (w) uvxxdx + j f2 (w)vxxdX} o 0 0

1 1 1

+ N IVxt 12 + f J f3 (w) Vttvxxdx + f J f~ (w) Wtvtvxxdx - a J Vltvxxdx o 0 0

1 1 1

+ J it (w) Utvxxdx + J f{ (w) Wtuvxxdx - J f~ (w) Wtvxxdx o 0 0

172 w. Fitzgibbon et al.

1

+N IVxtl 2 - J h (w) [NVt - Vxx + fh (w)Vt + 11 (w)u - aVI - h (w)]vxxdx o

1 1 1

+f J I~ (w) WtVtVxxdx - a J VltVxxdx + J II (w) Utvxxdx o 0 0

1 1

+ J I{ (w) Wtuvxxdx - J I~ (w) Wtvxxdx = O. o 0

Then taking the inner-product of the equation (25) with -1JVxx in L2 (0, 1), we get

(EVtt + NVt + fh (w) Vt - Vxx + 11 (w) U - aVI - h (w), -1JVxx) = -!tf1J (vx!' vx) - f1J Ivxtl 2 (27)

+1] Ivxx l2 ((N + fh (w)) Vt +!I (w) u - aVl - 12 (w), -1]Vxx )'

Summing up (26) and (27), we obtain

d 2 2 dt ii (t) + (N - f1J) Ivxtl + 1] Ivxxl ~ IK (tj u[B, Vb w)llvxxl, (28)

where

III 1

-f. J h (w) VtVxxdx + J aVlvxxdx - J !I (w) uvxxdx + J h (w) vxxdx, 000 0

and

K (t; U, VI, w) = h (w) [Nvt + fh (w)Vt + 11 (w)u - aVl - 12 (w)]

Global Dynamics 173

-17 [(N + ffa (W)) Vt + II (w) U - aVI - 12 (w)].

We have (19) and a corresponding decay estimate for the component VI. Besides,

IWt (t, x)1 ~ Ihl (u (t,x))llw (t,x)I+lh2 (u (t,x))1 ~ Ch (1 + Cw ), Vt ~ 0, x E [0,1],

where Ch > 0 is the uniform bound for both hI and h2• Hence, there is a constant CJ( (r) > 0, which depends on r > 0, such that IK (tj U, Vb w)1 ~ CJ(, for any (uo, Ul, wo) E BTl with BT a bounded ball in E centered at the origin and with radius r. Therefore,

(29)

if." > 0 is chosen to satisfy

N o < ." ~ -2 . (30) fO

Moreover, for small 6 such that 0 < 6 ~ min {N/fo, 17/2}, we have

~ [N IVxtl 2 +." Ivxx l2] - 6TI (t) = ~ (N - f6) IVxtl2 + ~ (17 - 6) Ivxx l2 - &." (VXh Vx) 111 1

+6f J 13 (W) Vtvxxdx - 6 J aVlvxxdx + 6 J It (w) uvxxdx - 6 J 12 (w) vxxdx 000 0

;::: -a IL (tj U, VI, w)llvxxl, (31)

where L(tj u, Vb W) = ffa(W)Vt - aVI + Idw)u - I2(w) - &",Vt. Similarly, there is a constant CL (r) > 0, which depends on r > 0, such that IL (tj U, Vb w)1 ~ CL (r), for any (uo, Ut, wo) E BT, with BT a bounded ball in E centered at the origin and with radius r. From (29) and (31) it follows that

d 0 (r dt II (t) + Sll (t) + ~ Ivxx l2 :'5 SOL(r) Ivxxl + K TJ r

< ?ll 12 + S20L(r)2 + OK (r)2 - 4 Vxx TJ TJ'

or

d 1 [ 2 2 2] dtll (t)+Sll (t) :'5 ~ OK (r) + 15 OL (r) ,for t ~ 0 and any (uo, Ut, wo) E Br •

(32) Because II (0) = 0, it follows that

II (t) :'5 SlTJ [OK (r)2 + S20L (r)2] , t ~ o.

Consequently, we have

(33)

which implies, by the same technique of using Young's inequality to handle the integrals in (33), that

for any (uo, UI, wo) E Br C E. Finally, we treat W2 (t, x) which satisfies the following equation:

d dt W2 = -hI (U)W2 + h2 (u), W2 (0) = 0, t ~ o. (35)

By taking derivatives and inner-product with W2x, we get

Global Dynamics 175

I

~1t IW2xl2 + b IW2xl2 :s; ~1t IW2l + [ hI (U) IW2l dx (36)

:s; I(h~ (u) UxW2 + h~ (u) ux, w2x)1 :s; % IW2l + Q (r),

where Q (r) is a constant depending on r, such that for any (uo, UI, wo) E Br, and for t ~ 0, h~ (u) lux llw21 + Ih; (u)lluxl :s; Q (r). Thus there is a constant Q* (r) > 0, such that

(37)

for t ~ 0 and any (uo, UI, wo) E Br . Combining (30) and (33), we conclude that

Ut>o 5(2 (t) Br C a bounded ball in H2 (0,1) X HI (0,1) X HI (0, 1), - centered at the origin and with radius [C* (r) + Q* (r)]1/2 .

(38) By the Sobolev imbedding theorem, H2 (0,1) X HI (0, 1) X HI (0, 1) is embedded in E compactly. Therefore, the component family of operators 5(2 (t) satisfies the condition (ii) in the basic theorem. This completes the proof of Theorem 3. 2

References

[1] J.W. Evans, "Nerve axon equations I: linear approximations," Indiana Univ. Math. J., 21 (1972),877-885.

[2] J.W. Evans, "Nerve axon equations II: stability at rest," Indiana Univ. Math. J., 22 (1972), 75-90.

[3] J.W. Evans, "Nerve axon equations III: stability of the nerve impulse," Indiana Univ. Math. J., 22 (1972),577-593.

[4] J.W. Evans and N.A. Shenk, "Solutions to axon equations," Biophys. J., 10 (1970), 1090-1101.

[5] W.E. Fitzgibbon and M.E. Parrott, "Convergence of singularly perturbed Hodgkin-Huxley systems," Nonlinear Anal., TMA, to appear.


[6] J.K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, R.I., 1988.

[7] O.A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge Univ. Press, Cambridge, England, 1991.

[8] H.M. Lieberstein, "On the Hodgkin-Huxley partial differential equation," Math. Biosci., 1 (1967), 45-69.

[9] M. Marion, "Finite dimensional at tractors to partly dissipative reactiondiffusion systems," SIAM J. Math. Anal., 20 (1989), 816-844.

[10] B. Najman, "Time singular limit of semi linear wave equations with damping," preprint 1990.

[11] J. Smoller, Shock Waves and Reaction-Diffusion Equations, SpringerVerlag, New York, 1983.

[12] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer- Verlag, New York, 1988.

W. Fitzgibbon Department of Mathematics University of Houston Houston, Texas 77004

M. Parrott Department of Mathematics University of South Florida Tampa, Florida 33620-5700

Y. You Department of Mathematics University of South Florida Tampa, Florida 33620-5700

On strongly elliptic differential operators on Ll(JRn)

Matthias Hieber

1. Introduction

Let A := LIQI=m aQDQ be a homogeneous differential operator with

constant coefficients a Q • Then A is called strongly elliptic if

Rea(~) > 0 for ~ E JRn\{O}.

Here a is the symbol of A defined by a(~) := LIQI=m aQ(i~)Q. We

remark first that the order m of A is necessarily even, due to the

fact that a( -~) = (-l)ma(~) for all ~ E JRn. Second, consider the

LP-realization Ap of A (1 ::; p ::; co) given by

(1.1) D(Ap):= {I E LP(JRn);.:F-1(aj) E LP(JRn)}

ApI := .:F-1(aj) for all IE D(Ap).

and

Here :F denotes the Fourier transform in S', the space of tempered

distributions. Then it is well known that the operator -Ap gener

ates a holomorphic Go-semigroup on LP(JRn) whenever 1 < p < co.

Moreover, this assertion remains valid for the variable coefficient

operator LIQI~m aCt(x )DCl, provided that A is uniformly strongly el

liptic and aCt E L= for all lad::; m and aCt E BUG for all lal = m.

The crucial argument in the proof of the results mentioned above is

Mihlin's multiplier theorem (cf.[S;p.96]), which in particular implies

that D( Ap) = H;(TR n). Note that in the case p = 1 neither Mihlin's

theorem nor the above characterization of D( Ap) remain valid. 177

178 M. Hieber

Assuming the Holder continuity of the coefficients, Guidetti

[Gu2] recently proved that operators of this kind generate holomor

phic semigroups also on Ll(JRn ). For similar results in this direction,

we refer to [Gul] and [R].

The aim of this paper is to show that if the coefficients of the

principal part are constant, i.e. if aa E (C for lal = m, then one can

give an easy proof of the result mentioned above.

2. Estimates on the principal part

Let A be strongly elliptic. In order to prove that -AI generates

a holomorphic semigroup on Ll(JRn ), we make use of the following

simple sufficient criterion for a function to belong to MI. Here M 1

denotes the Banach space defined by

where

For € > 0, we denote by Me the Banach space

where j = min{k E IN, k > i} and

Lemma 2.1 ([Hi;Lemma 3.2]. Let c > O. Then Me y FLl(JRn) y

MI'

Elliptic Differential Operators 179

We remark that if a E Me, then the mapping Ta : f 1-+ F-I(aj) is a

bounded, linear operator on LI(lRn) with IITa II ~ ClialiMc for some

C > o. Lemma 2.1 implies in particular that the spectrum of -AI is

contained in a sector of the form

Se := {z E <C; largz - 71"1 ~ (:J}

for some (:J satisfying 0 ~ (:J < f. To be more precise, (:J can be

calculated to be (:J = maxlel=I arctg~:f~~. In particular, if a Q E lR, then So = lR_ U {a}.

Second, Lemma 2.1 is the key tool in the proof of the following

result (see also [Gu2] and [R;Ch.V]). In our proof we follow an idea

of Arendt [A].

Proposition 2.2. If A is strongly elliptic, tben -AI generates a

bounded, bolomorpbic Co-semigroup on LI(lRn).

Proof. In view of [N;Thm.AII 1.14], we first prove that

sup 11-\(-\ + Ad-III < 00. Re>.>O

To this end, let -\ = eit/> for some . : lRn ~ <C is defined by r>.(O = -\(-\ + a(O)-I. We claim first that r>. E MI with norm bound

independent of -\. Differentiating r>. , we obtain

where the constant C is independent of -\. Hence, Lemma 2.1 implies

that

sup 11-\(-\ + Ad-III < 00. Re>'>o,I>'I=I

180 M. Hieber

Finally, let A E (!; such that ReA > 0 and note that • ...L (0 = P·I

I~I (I~I + a(O)-I. Define s,\ by s,\(O := • Tti( ~). Since Ml is invariant under homothetic maps of JRn (cf.[Ho;Thm.1.13]), we

conclude that s,\ EMI and that IIs'\IIMl = II • ...LIIM1· Since s,\(O = IAI

A( A + a( 0) -1 for all A E (!; with ReA> 0, the proof is complete. 0

Remark. The proof shows that the assertion of the above proposi

tion remains true also for the spaces Co(JRn) and BUC(JRn ).

Note that the domain of Al is different from the Sobolev space

Wr or the Bessel-potential space Hrn. However, D( AI) can be

sandwiched by suitable function spaces. To be more precise, for

S E JR define P : JRn -4 JR by J8(0 := (1 + leI2)8/2. Then, following

the standard notation (cf.[BL]), we set for m E IN' U {O} and S E JR

W1m := {f E SI(JRn); na f E Ll(JRn) for all lal ~ m} Ht:= {f E S'(JRn);F-1(Pj) E Ll(JRn)}.

Lemma 2.3. Let -AI be a strongly elliptic operator on Ll(JRn) of

order 2m. If 0 2m - 1, then Hi ~ w;m-I.

Proof. In order to prove that D( Ad ~ H;m-8, it suffices to show

that the function e ~ j2m-8(0.,\(e) belongs to MI. Here.,\ is

defined by .,\(0 := (A + a(O)-1 for some A E (!; with ReA> O.

But this follows easily from Lemma 2.1. On the other hand, let

u E H;m+8. Again Lemma 2.1 implies that the function 9 given

by e ~ ea J-(2m+8) (0 belongs to M 1 for all a with lal = 2m.

Hence F-l(gu) E Ll and therefore u E D(At). The last assertion is

similarly easy to prove. 0

Remark. We are aware of the stronger result

due to Guidetti [Gu1]. Here B;q denotes the Besov space defined as

in [BL] or [T]. However, the easy accessible assertion of Lemma 2.3

is sufficient for our purposes.

3. The main result

Consider now a differential operator with variable coefficients of the

form

lal=2m lal9m- 1

where aa E <C and ba E LOO. We assume the principal part AH :=

Llal=2m aa Da of A to be strongly elliptic. Moreover, setting aH :

JRn --+ <C,aH(O:= Llal=2m aa (ioa, we define the Ll-realization of

A by

D(Ad := {f E LI(JRn); F-I(aH [) E LI(JRn)} and

Alf := Af for all f E D(Ad·

Theorem 3.1. Let AH be a strongly elliptic operator of order 2m

and let B be given by B := Llal9m- 1 baDa, where ba E LOO. Then

-AI generates a holomorphic Co-semigroup on LI(JRn).

Proof. Notice first that by Proposition 2.2, the operator -Af is the generator of a holomorphic semigroup (TH (t))t>o on LI(JRn).

Obviously, IIBuill :S const.llullw2m-1. Let 8,0 E (0,1). Then, by 1

real interpolation (cf. [BL;Thm.6.2.4]), we obtain

H 2m - o '--+ (Ll H 2m - O) _ B O(2m-o) H O(2m-o) 1 '1 0,1 - 11 '--+ 1 ,

182 M. Hieber

where Bf~2m-8) denotes the Besov space defined as in [BL]. More

over, for every [, > 0, there exists a C~ > 0 s,uch that

Furthermore, by Lemma 2.3, we conclude that H~(2m-8) '-t w;m-1 provided

e > ;:=~. Therefore, by choosing [, small enough, it follows from

Lemma 2.3 that

for all u E H;m-8, all [,' > 0 and suitable C:. Hence B is relatively

Af"-bounded and the claim follows from the classical perturbation

theorem for holomorphic semigroups (cf.[Go;Thm.I.6.6]). 0

References

[A] W. Arendt, Linear Evolution Equations. Lecture Notes, Univerity of Zurich, (1991).

[BLl J. Bergh, J. Lofstrom, Interpolation Spaces. Springer-Verlag Berlin, Heidelberg, New York, (1976).

[Go] J.A. Goldstein, Semigroups of Linear Operators and Applications. Oxford University Press, (1985).

[Gu1] D. Guidetti, On interpolation with boundary conditions. Math. Z. 207 (1991), 439-460.

[Gu2] D. Guidetti, On elliptic systems in L1. preprint 1991.

[Hi] M. Hieber, Integrated semigroups and differential operators in LP spaces. Math. Ann. 291 (1991), 1-16.

[Ho] L. Hormander, Estimates for translation invariant operators in LP spaces. Acta Math. 104 (1960), 93-140.


[N] R. Nagel (ed.), One-parameter Semigroups of Positive Operators. Lecture Notes in Math. 1184, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, (1986).

[R] D.W. Robinson, Elliptic operators and Lie groups. Clarendon Press, Oxford, New York, Tokyo (1991).

[S] E. M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, New Jersey, 1970.

[T] H. Triebel, Theory of Function Spaces. Birkhauser Verlag, Basel, Boston, Stuttgart (1983).

Mathematisches Institut Universitat Ziirich Ramistrasse 74 CH-8001 Ziirich Switzerland

Stability and local invariant manifolds in fully nonlinear parabolic equations

Alessandra Lunardi 1

1. Introduction

We study the asymptotic behavior for small initial data of the solutions of a class of evolution equations in general Banach space X:

u'(t) = Au(t) + F(u(t)), t ~ 0,

u(O) = uo.

(1)

(2)

Here A: D(A) C X 1---+ X is a sectorial operator, and F: DA (() + 1,00) 1---+

DA((),oo) is a nonlinear sufficiently smooth function, such that F(O) = 0, F' (0) = o. We recall that D A ( (), 00) is the usual real interpolation space between X and D(A), and DA (() + 1,00) = {x E D(A): Ax E DA((),oo)} is the domain of the part of A in D A ((), 00 ).

In particular, we state existence and regularity results for initial data in DA (() + 1,00), we prove that the principle of linearized stability holds, and we construct local stable, unstable, and center manifolds.

Such a type of nonlinearity arises in the study of fully nonlinear parabolic equations and systems. As a basic example, we consider

{ ~t(t,X) = ~u + cu + f(~,DU,D2U), t ~ 0, x E n, (3)

aujav = 0, t ~ 0, x E an, where n is a bounded open set in R n with regular boundary an, ~ is the Laplace operator, v is the exterior normal vector to an, and (u, p, q) 1---+

f( u, p, q) is defined and twice continuously differentiable in a neighborhood of 0 in R x Rn x Rn2

, f(O) = 0, Df(O) = o. 1 Partially supported by GN AFA of CNR.

185


186 A. Lunardi

As usual in fully nonlinear equations, we choose X = C(O). Then the realization A of L:.. + cI with homogeneous boundary condition in X is sectorial. The domain D(A) is defined by

D(A) = {cp E n W 2,P(O) : L:..cp E X, f)cp/f)v = 0 in f)O}, p~l

and it consists of twice continuously differentiable functions only if the space dimension n is equal to 1. Therefore, the composition F( cp) = f( cp, Dcp, D2cp) is not well defined on D (A) if n > 1, but it is well defined on the space

for every B E ]0,1/2[' and it has values in D A(B, 00) = C2B (O). \Ve use some results of [7], where local existence and regularity for small

initial data Uo E D A (e + 1,00) were proved. In particular, the solution of (1)-(2) is represented by the variation of constants formula

l1(t) = etAuo + l e(t-s)A F(u(s))ds, t::::: 0, (4)

which lets us adapt to our situation the Lyapunov-Perron approach for the construction of invariant manifolds. The difficulties arising from the fully nonlinear character of (1) are overcome by using optimal regularity results (and the corresponding estimates) for the linearized problem

v'(t) = Av(t) + f(t), both for forward and for backwaTd solutions.

The construction of the center-unstable manifold, in the general case, relies on the assumption that the elements of the spectrum of A with nonnegative real part are a finite number of eigenvalues, with finite algebraic multiplicity. If this assumption fails to be satisfied - for instance, if the set 0 in example (3) is unbounded - other techniques have been used in the study of the stability of the null solution: see [7] and the references quoted therein.

The results about problem (1) are similar to the ones of [3], where the continuous interpolation spaces D A (B), D A ((} + 1) were used instead of D A (B, 00), D A( e + 1, 00). We recall that D A( B) is the closure of D(A) in D A(B, 00), and

Local Invariant Manifolds 187

DA(B + 1) = {x E D(A) : Ax E DA(B)} is the domain of the part of A in DA(B). In the applications to parabolic problems such as (3), DA(B) turns out to be the space h20(0) of the little-Holder continuous functions of exponent 2B, which is the closure of CCXl(O) in C20(0). Correspondingly, DA(B+1) = {<p E C2(0) : DiN E h20(0), o<p/ov = o}.

It seems useful to develop a theory which avoids the little-Holder spaces and employs only the more familiar Holder spaces. Moreover, we do not follow closely the proofs of [3] but we simplify and clarify them. In particular, we find new existence in the large results (Proposition 9).

The abstract results are applied to the study of the behavior of the solutions of problems of the type (3) (a.lso with Dirichlet boundary condition) for small initial data. Concerning (3), for c < ° the null solution is exponentially asymptotically stable in C2+28 ("IT), if c > ° the null solution is unstable, and we prove the existence of an infinite dimensional stable manifold and of a finite dimensional unstable or center-unstable manifold. For c = 0, we show that the subset of C2+28 ("IT) consisting of the functions with zero normal derivative and zero mean value is a local center manifold. In particular, this fact reduces the problem of stability for (3) to the problem of stability for the ordinary differential equation u' = J( u, 0, 0). Therefore, sufficient conditions for stability are easily found: if

and either k is even, or k is odd and ak J / auk(O, 0, 0) > 0, then the null solution is unstable; if k is odd and ak J / auk (0,0,0) < 0, then the null solution is stable in C2+28(0).

2. The principle of linearized stability, the stable and the unstable manifolds

Let X be a Banach space with norm II . II, and let A : D(A) C X f---t X be a linear operator such that there are constants w E R, B E ]7r /2, 7r[, M > ° satisfying

M p(A) =:J S = {,\ f. w: larg(A - w)1 < B}, IIR(,\, A)IIL(x) ::; IA _ wi VA E S.

(5)

188 A. Lunardi

Then A generates an analytic semigroup etA in X (see [11]). The domain D(A) is endowed with the graph norm IlxIID(A) = Ilxll + IIAxll.

The interpolation spaces D A ((), 00 ), D A (() + 1,00) are defined, for 0 < () < 1, by

{ DA((),oo) = {x EX: IxIDA(O,oo) = sUPo<t:9lltl-OAetAxll < oo},

IIXIlDA(O,oo) = IIxll + IxivA(O,OO);

{ DA(() + 1,00) = {x E D(A) : Ax E DA((),oo)},

IlxIIDA(B+l,oo) = IlxIID(A) + IAxIDA(O,oo)'

A local existence and uniqueness theorem for the solution of (1 )-(2) has been shown in [7J.

Theorem 1 Let A : D(A) C X 1---+ X satisfy (5). Let n be a neighborhood of a in DA(() + 1,00), with a < () < 1, and let F : n 1---+ DA((),oo) be a continuously diffel'entiable function. Then for every T > a there are R = R(T), Ro = Ro(T), M = M(T) > a such that if IluoIIDA(O+l,oo) ~ R, then problem (1)-(2) has a solution u E C(]O, TJ; DA(()+l, 00)) n CO([O, T]; D(A)) n Cl(]O,TJ;DA(()'oo)), such that

sup Ilu(t)IIDA(B+l,oo) + lIullc8([o,T);D(A)) ~ MlluoIIDA(II+l,oo)' (6) 09~T

u is the unique solution of (1)-(2) such that sUPO~t~T Ilu(t)IIDA(B+l,oo) ~ Ro.

As a corollary, a continuation result follows.

Corollary 2 Set R = sUPT>O R(T), where R(T) is given by Theorem 1. Let u be a solution of (1) and set T = sup{t > a : u E C(]O, tJ; DA(() + 1,00)) solves {1J in JO, tl}. If Ilu(t)IIDA(B+l,oo) < R for every t < T, then T = +00.

Theorem 1 a.nd its corolla.ry will be used later. The results concerning the principle of linearized stability are independent of them. We begin with the stability theorem, which looks similar to the linearized stability theorem for semilinear ([5]) or quasi linear equations ([4], [6], [10]).

Theorem 3 Let A : D(A) C X t---7 X be a linear operator satisfying (5), such that

sup{ Re).: ). E a(A)} = -Wo < O. (7)

Then for even) w E ]O,wo[ there are r, M such that if IluoIIDA(8+1,00) :::; r, then the solution u of (1)-(2) is defined in [0, +00[, and

The proof of Theorem 3 relies on a similar result for the linear case. Since we deal with exponentially decaying functions, we introduce some weighted spaces. If Y is any Banach space and w 2:: 0, we set

{u E C(]O, +00[; Y) : t t---7 ewtu(t) E Loo(O, +00; Y)},

lIullc.,()o,+oo[;y)

Proposition 4 Assume that A satisfies (5) and (7). Let 0 < 0 < 1. Then for every Uo E DA(O + 1,00) and f E Cw(]O, +00[; DA(O, 00)), the solution of problem

l/(t) = Av(t) + f(t), t 2:: 0; v(O) = uo, (8)

belongs to Cw(]O, +00[; D A (0 + 1,00)) n Cl (]O, +00[; D A( 0,00 )), and there is C = C(w) > ° , not depending on f, Uo, such that

The proof of Proposition 4 follows the same arguments of [9], Proposition 3.10, where the nonautonomous periodic case A = A(t) is treated. The details of this (simpler) case will appear in [8].

Proof of Theorem 3 - Once Proposition 4 has been established, the proof of Theorem 3 is quite similar to the proof of local existence and uniqueness

190 A. Lunardi

given in [7]. Indeed, every regular solution of (1 )-(2) in [0, +oo[ is a fixed point of the operator f defined by

Using Theorem 4, it is easy to see that if Ro is sufficiently small, then f is a 1/2-contraction on the ball B(O,Ro) C Cw(]O,+oo[jDA(O + 1,00)). Setting

we get Ilfullcw(]o,+oo[;DA(II+l,oo)) ::; Mo,wlluoIIDA(8+1,oo) + Ro/2, so that f maps B(O, Ro) into itself provided IluoIIDA(O+l,OO) ::; r = (2Mo,w)-1 Ro, in which case it has a unique fixed point u in B(O, Ro), which is the solution of (1)-(2) enjoying the claimed properties. •

For the instability result, the assumption corresponding to (7) is the following:

a+(A) = a(A) n {A E C : ReA> O} =J 0, inf{ Re A: A E a+(A)} = w+ > 0. (10)

If (10) holds, one can define the projection

1 1 -1 P = -2' (A - A) dA, 1n "I

where'Y is any regular curve surrounding a+(A), lying in the half plane { Re A > OJ, oriented counterclockwise. Then P(X) C D(An) for every n E N. We use again spaces of exponentially decaying functions, defined as follows: if Y is any Banach space and w > ° we set

Cw(] - 00,0]; Y) {u E C(] - 00, O]j Y) : t I-t ewtu(t) E LOO( -00, OJ Y)},

II U IICw (]-oo,O];Y)

Theorem 5 Let A, F satisfy the assumptions of Theorem 1 and assume that (10) holds. Then there exist nontrivial backward solutions u E C(]oo,O]jDA(B + 1,00)) of (1), going to ° as t goes to -00. In particular, the

null solution is unstable. More precisely, fixed any w E ]0, w+[ there are 1'0,

rlJ M > ° and a Lipschitz continuous function

<p: B(O,ro) C P(X) 1-+ B(O,rd C (I - P)(DA(O + 1,00)),

differentiable at ° 'With <p' (0) = 0, such that for every Uo belonging to the graph of <p problem (1) has a unique back'Ward solution u E Cw(] - 00,0]; D A (0 + 1,00)) 'With Ilullc..,(]-00,0j;DA(9+1,00)) ::; M, and, conversely, ifproblem (1) has a back'Ward solution u E Cw(] - oo,O];DA(O+ 1,00)) 'With norm less than M, then u(O) belongs to the graph of <p.

The graph of <p is called unstable manifold. As in the stability case, the proof of Theorem 5 relies on a result about backward solutions of linear problems, whose proof is quite similar to Proposition 3.11(ii) of [9], and is omitted.

Proposition 6 Assume that A satisfies (5) and (10). Let ° < 0 < 1, and fix w E jO,w+[. Then for every f E Cw(] - 00, OJ; D A(O, 00 )), all the solutions z E Cw(] - 00, OJ; D A(O + 1,00)) of problem

Zl(t) = Az(t) + f(t), t::; 0,

are given by the repl'esentation formula

z(t)=etAx+ (te(t-S}APf(s)ds+jt e(t-s}A(I-P)f(s)ds, t::;O, Jo -00

'Where x is any element of P(X). MOl'eover, they are continuously differentiable 'With values in DA(O, 00), and there is C > ° , not depending on j, x, such that

II Z llcw(]-00,O];DA(8+1,00}} ::; C(llxll + IIfllcw(]-00,0j;DA(8,00}})' (11)

Proof of Theorem 5 - Fixed x E P(X), we look for a fixed point of the operator A defined by

192 A. Lunardi

in a small ball B(O,M) C Cw(] - oo,O];DA(B + 1,(0)), with ° < w < w+. Using estimate (11) it is not difficult to see that Ax,M is a 1/2-contraction in B(O, M) if M is sufficiently small; moreover it maps B(O, M) into itself provided jjxjj ~ ro = M/2C (C is the constant in estimate (11)). The first part of the statement follows. Concerning the second part, we set

<p(x) = (I - P)u(O) = LOoo e-sA(I - P)F(u(s))ds,

where u is the fixed point ofAx,M' Since the mapping (x, u) I---T Au is Lipschitz continuous in B(O,ro) x B(O,M) C P(X) x Cw(] - oo,O];DA(B + 1,(0)), by the contraction principle depending on a parameter, also <p is Lipschitz continuous. Moreover, since Ax,M is a 1/2- contraction, it holds

jjujjC..,(]-oo,O];DA (B+l,oo)) ~ 2Cjjxjj,

so that

Letting M -+ 0, since F'(O) = ° we find <p'(0) = 0. •

In the saddle point case, that is when the spectrum of A does not intersect the imaginary axis, also a local stable manifold may be easily constructed. We need a linear result similar to the one of Proposition 6, for whose proof we refer to [9, Proposition 3.10] or to [8].

Proposition 7 Assume that A satisfies (5) and that

a(A) n iR = 0. (12)

Let ° < e < 1, and fi:c w E ]0, w_ [, 'Where -w_ = sup{ Re).: ). E a(A), Re ). < O}. Then for every f E Cw(]O,+oo[;DA(B,oo)), all the solutions of problem

v'(t) = Av(t) + f(t), t ~ 0,

belonging to Cw(]O, +00[; D A (0 + 1,(0)), are given by the representation formula

r 1+00 v(t) = etA.]; + Jo e(t-s jA(1_ P)f(s)ds - t e(t-s)Apf(s)ds, t ~ 0,

where x is any element of (I - P)(DA (() + 1,00)). Moreover, they are continuously differentiable in ]0, +00 [ with values in D A ((), 00 ), and there is C = C(w) > ° , not depending on f, x, such that

Using proposition 7 and arguing as in the proof of Theorem 5, one gets

Theorem 8 Let A, F satisfy the assumptions of Theorem 1 and in addition assume that (12) holds. Then, fixed any w E ]O,w_[ there are ro, rl, M> ° and a Lipschitz continuous function

7/;: B(O,ro) c (I - P)(DA(O + 1,00)) 14 B(O,rd c P(X),

differentiable at ° with 7/;'(0) = 0, such that for every Uo belonging to the graph of 7/; problem (1) has a unique solution U E Cw(]O, +00[; D A( 0 + 1,00)) such that Ilullc",(]o,+oo[;DA(8+1,oo)) ::::: M, and, conversely, if problem (1) has a solution U E Cw(]O, +00[; D A( () + 1,00)) such that Ilullc",(]o,+oo[;DA(8+1,oo)) :::::

M, then u(O) belongs to the graph of '1/'.

The graph of 7/; is called stable manifold.

3. The center and center-unstable manifolds

Throughout the section we shall assume that

{the set 0"+ (A) = P. E O"(A) : Re A 2:: O} consists of a finite number of isolated eigenvalues with finite algebraic multiplicity.

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We set as before w_ = - sup{ Re A: A E O"(A), Re A < O}, and P = 2~i J../A - At1dA, where f is any regular curve surrounding O"+(A) = P. E O"(A), Re A 2:: O}, lying in the half plane { Re A > -wo}, oriented counterclockwise.

194 A. Lunardi

Then problem (1) is equivalent to the system

{ x'(t) = A+x(t) + PF(x(t) + y(t)), t:2: 0,

(15) y'(t) = A_y(t) + (I - P)F(x(t) + y(t)), t:2: 0,

with x(t) = Pu(t), y(t) = (I - P)u(t), A+ = AIP(X) : P(X) f-t P(X), A_ = AI(I-P)(DA(B+l,oo)) : (I - P)(DA(O + 1,(0)) f-t (I - P)(DA(O, (0)). As usual, we modify F by introducing a smooth cutoff function p : P(X) f-t R,

such that ° ::; p(x) ::; 1, p(x) = 1 if Ilxll ::; 1/2, p(x) = ° if Ilxll :2: 1, and for small p > ° we consider the system

where

{ x'(t) = Aox(t) + f(x(t),y(t)),

y'(t) = A_y(t) + g(x(t), y(t)),

t :2: 0,

t :2: 0,

f(x,y) = PF(p(x/r)x + y), g(x,y) = (I - P)F(p(x/r)x + y).

(16)

System (16) coincides with (15) if Ilx:(t)11 ::; 1'/2. In particular, they are equivalent as far as stability of the null solution is concerned.

Theorem 1 and Corollary 2 may be applied to problem (16), getting local existence for small initial data, and existence in the large provided the solution remains small enough. In fact, while it is easy to find an a priori estimate on Ily(t)IIDA(e+l,CXl) for l' small (see Proposition 9 below), in general x(t) is not necessarily bounded (for instance, in the case where f == ° and II etA PIIL(X) is not bounded there are arbitrarily small Xo such that x(t) is not bounded). Therefore, Corollary 2 is of no use here. However, due to the truncation in f and g, we can show that if r and the initial data are small enough, then the solution of (16) exists in the large.

Proposition 9 There are 1'0, Co = Co(r), such that for l' ::; 1'0 and IIxoll + 11Y01IDA (B+1,CXl) ::; Co(r), problem (16) has a solution (x, y) E C([O, +00[; P(X)) x C(]O,+oo[;DA(O+l,oo)) n LCXl(O,+oo;DA(O+I,oo)) n C([O,+oo[;D(A)). The solution is unique in the class of the functions (x, y) enjoying the above properties of regu.larity and such that Ily(t)IIDA (B+l,oo) ::; r.


Proof - The proof is in two steps. First we show that for any fixed T > 0, system (16) has a solution in [0, T] provided r is small and IiYoIIDA(8+1,00) ~ r/2M, where

M = sup lIetA-IIL((I-P)(DA(8+1,00)), t>o

no matter how Ilxoll is large. Second, we prove that liy(t)IiDA(8+1,00) ~ r/2Mfor every t in the maximal interval of existence, provided IIxoll and IiYoIIDA(8+1,00) are small enough.

For every r > 0, denote by L(r) the maximum between the Lipschitz constants of J and 9 over P(X) x B(O, r) C (I -P)(D A(O+l, 00)) (obviously, P(X) is endowed with the norm of X and (I - P)(DA(O + 1, 00)) is endowed with the norm of DA(O + 1,00)). Then

lim L(r) = 0. r->O

Let moreover C_ be the constant of estimate (9), with A replaced by A_ and w = 0, and let C+ = C+(T) be such that

lifot e(t-s)A+ J(s)dsll ~ C+IIJlluX'(O,T;X), ° ~ t ~ T,

for every J E Loo(O, T; P(X)). Any solution of (16) is a fixed point of the operator r defined by

r(x,y)(t) = (etA+xo + fote(t-S)A+ J(x(s),y(s))ds,

Ifr is small enough, r is well defined on the set Y = B(etA+xo,r) x B(O,r) C C([O, T]; P(X)) x C(]O, T]; (I - P)(DA(O + 1,00)) n Loo(O, T; (DA(O + 1,00)). Y is endowed with the product norm II(x,y)lly = IIxIlLOO(O,T;X) + IlyIlLOO(O,T;D A(8+1,00))·

For (x,y), (x,y) in Ywehave IIJ(x(s),y(s))-J(x(s),y(s))1I ~ Lr (1Ix(s)x(s)1I + Ily(s) - y(s)IIDA(8+1.00)), Ilg(x(s),y(s)) - g(x(s),y(s))IIDA(8,00) ~ Lr (lix(s) - x(s)11 + Ily(s) - y(s)IIDA (8+1,00)). Therefore, r is a 1/2-contraction provided

196 A. Lunardi

Moreover, setting M(r) = supxEP(X).IIYIIDA(e+l.oo)~T Ilf(x, y)II+lIg(x, y)IIDA(O,OO)' we have

limM(r)/r = O. r-O

For (x, y) E Y we get

In particular, r maps Y into itselfifr is so small that (C++C_)M(r)/r ~ 1/2 and IIYoIIDA(B+l.oo) S r/2M. So, if r is small enough and IIYoIIDA(B+1,oo) S r/2M, problem (16) has a unique solution in Y. Indeed, it is the unique

solution such that Ily(t)IIDA(B+l.00) Sr. Let us prove now that Ily(t)IIDA(9+1.oo) S r/2M for every t in the interval

of existence if the initial data are small enough. By Theorem 1, there is

C(r) such that if Ilxoll + IIYoIIDA(9+1.oo) S C(r), then the solution of (16) is defined at least in [0,1], and Ily(t)IIDA(9+1.00) ~ r/4M for 0 ~ t ~ 1. Let 7 = inf{t > 0 : y(t) exists and lIy(t)IIDA(O+1.oo) ~ r/2M}. Since 7 ~ 1, then y is well defined and continuous at t = 7 with values in DA (()+1,00), so that

which is impossible if r is so small that C_M(r) ~ r /4M and IIYoIlDA(B+l.oo) ~ r/4M. For such values ofr and ofthe norms ofthe initial data, lIy(t)IIDA (B+1.oo) remains bounded by r /2M, as far as it exists. •

We shall state the existence of a finite dimensional invariant manifold Vc for system (16), provided r is sufficiently small. Then we shall see that such a manifold attracts exponentially all the orbits starting from an initial datum sufficiently close to it. As a last step, we shall see that the null solution of (16) is stable, asymptotically stable, or unstable, if and only if it is stable, asymptotically stable, or unstable, with respect to the restriction of the flow

to Vc'

Theorem 10 Let A satisfy (5) and (14). Then there exists ro > 0 "Such that for r S ro there is a Lipschitz continuous function <p : P(X) 1---7 (I -P)(D A(() + 1,(0)) such the graph of <p is invariant for system (16). If in


addition F is k times continuously differentiable, then cp E Ck- 1 , cpk-l is Lipschitz continuous, and

cp'(x)(Aox + f(x,cp(x))) = A_cp(x) + g(x,cp(x)), x E P(X). (17)

The proof is quite similar to the one of Theorems 3.1 and 3.2 of [3], to which we refer the interested reader. Here we recall only what will be used later: as usual (see e.g. [5, Chapter 6] for the semi linear case), cp is sought as a fixed point of the operator f defined by

(fcp)(x) = 1°00 e-SA-g(z(s)),cp(z(s))ds, x E P(X), (18)

where z = z( s; x, cp) is the solution of the finite dimensional system

{ z' = Aoz + f(z + cp(z)),

z(o) = x.

Fixed a > 0, one can show that f has a unique fixed point in the set

y = {cp: P(X) 1-+ (I - P)(DA(O + 1,00)): cp(o) = 0,

provided r is small enough.

(19)

In the case where the fixed point of f is continuously differentiable, equality (17) follows by replacing y = cp(x) in (16).

Let us prove a property of attractivity of the center manifold.

Proposition 11 Let F be twice continuously differentiable. For every w E ]O,w_[ there are r(w), M(w) such that if Ilxoll, II Yo II DA(ll+l,oo) are sufficiently smail, the solution of (16) with x(O) = xo, y{O) = Yo exists in the large and satisfies

198 A. Lunardi

Proof - We use the notation and the results of Proposition 9. So, let r ~ ro and Ilxoll + IIYoIIDA(B+l,oo) ~ Co(r), so that the solution of (16) exists in the large, and Ily(t)IIDA(O+l,oo) :S r. Let v(t) = y(t) - ip(x(t)). Since ip satisfies (17), then

v'(t) A_v(t) + g(x(t),y(t)) - g(x(t),ip(x(t)))

-ip'(x(t))(J(x(t), y(t)) - f(x(t), ip(x(t)))

A_v(t) + G(x(t),y(t)), t ~ 0.

Recalling that. IIG(x(s),y(s))IIDA(B,oo) ~ L(r)(l + a)llv(s)IIDA(O+l,ooj. from Proposition 4 with A replaced by A_, we get

where C = C(w). Taking r so small that CL(r)(l + a) ~ 1/2, the statement follows, wit.h M(w) = 2C. •

Using the technique of [3], one can see that the result of Proposition 11 holds uncler t.he mere assumption that F is continuously differentiable, but the proof is much longer.

Once exponential attractivity of Vc is established, one can prove that it is asymptotically stable with asymptotic phase, in the sense of next proposition. Again, the proof follows closely the proof of Theorem 3.3 of [3]; it is sufficient to replace everywhere DA (()) by DA((),oo) and DA(() + 1) by DA(() + 1,(0).

Proposition 12 For every w E ]O,w_[ there is C(w) > ° such that if Xo E P(X) and IIYoIIDA(B+l,oo) :S ro then there exists x E P(X), depending continuously on (:r:o, Yo), such that, setting z(t) = z(t; ip, x), it holds

Ilx(t) - z(t)11 + Ily(t) - ip(z(i))IIDA(O+l,oo) :S C(w)e-wtllyo - ip(xo)ll, t ~ 0. (21 )

Here ro is given by Pmposiiion 9, and z(t; ip, x) is the solution of (19).

From Proposition 12 we get. easily the following corollary.

Corollary 13 The null solution of (1) is stable (respectively, asymptotically stable, unstable) in D A (() + 1,(0) if and only if the null solution of the finite dimensional equation (19) is stable (respectively, asymptotically stable, unstable).

4. Applications to nonlinear parabolic problems

We consider a nonlinear evolution problem in [0, +oo[ xn, n being a bounded open set in Rn with C2+2B boundary an, ° < () < 1/2:

2 -

{ Ut=~u+cu+f(u,Du,D u), t~O, xEn,

au(t,x)/an = 0, t ~ 0, x E an, (22)

u(O,x) = uo(x), x E n, (23)

where ~ is the Laplace operator, c E R, f is a twice continuously differentiable function defined in a neighborhood of ° in R x Rn x Rn2

, with f(O) = 0, Df(O) = 0, and a/an denotes the normal derivative.

An example of an equation of the type (22) arising in Detonation Theory may be found in [2].

It is well known (see [12]) that the realization A of ~ + cI with homogeneous boundary condition in X = C(n) is sectorial, and that (see [1])

with equivalence of the respective norms. The Schauder Theorem implies

with equivalence of the norms. It follows that the mapping 1jJ 1---+ F( 1jJ),

(F(cp))(x) = f(1jJ(x),D1jJ(x),D21jJ(x)) (24)

is well defined in a neighborhood of ° in D A (() + 1, 00 ), with values in D A ((), 00 ). Moreover, F is continuously differentiable, as it is easy to check.

Therefore, setting u(t) = u(t, .), problem (22) may be seen as an evolution equation in X of the type (1), such that the assumptions of Theorem 1 are

200 A. Lunardi

satisfied. Applying Theorem 1, one proves local existence and uniqueness of a classical solution u of (22) for small initial data Uo E C2+29(O) such that Buo/ Bn = 0. The regularity properties of the solution are the following: Ut and Diju are continuous in [0, T] x 0, with T > 0, and they are 2{)Holder continuous with respect to the space variables, with Holder constant independent of t; D.u is ()- Holder continuous with respect to time, with Holder constant independent- of the space variables. Moreover, t 1--+ u(t,·) is continuous in ]0, T] with values in C 29+2 (O).

If we denote by {-An}nEN is the ordered sequence of the eigenvalues of D. with homogeneous Neumann boundary condition, then a(A) = {-An + C }nEN. In the case where c < 0, the spectrum of A is contained in the left complex ha.lfplane. Then the principle of linearized stability may be applied, and it gives existence in the large and exponential decay of the solution for small initial data. Precisely, since the first eigenvalue of A is c, then for every wE jO,c[ there are M(w), R(w) such that if Iluollc2B+2(O) ::; R(w), then

Ilu(t, ·)lbe+2(IT) ::; M(w)e-wt lluollc26+2(IT), t ~ 0.

In the case where c > 0, the assumptions of Theorem 5 are satisfied, so that the null solution is unstable, and there exists a finite dimensional Lipschitz continuous unstable manifold. If An + c#-O for every n EN, then ° is a saddle point: there exists also a Lipschitz continuous infinite dimensional stahle manifold.

In the critical case of stability c = 0, the part of the spectrum of A in the imaginary axis consists only of the simple eigenvalue 0, and the corresponding eigenspace is the set of the constant functions. The operator Ao is the null operator, and the projection P is given by

1 1 -P~jJ(() = 0 1jJ(y)dy, (E O. meas fl

If f = f(p,q) does not depend explicitly on u, then F vanishes on P(X), and this fact lets one decouple system (15), which becomes (with the notation of Section 3)

{ x'(t) = P F(y(t)), t ~ 0,

y'(t) = A_y(t) + (/ - P)F(y(t)), t ~ 0. (25)

The second equation satisfies the assumptions of the principle of linearized stability, with X replaced of course by (1 - P)(X), because the spectrum of A_ consists of the negative eigenvalues -An, n ~ 1. By applying Theorem 3, we find that for every w E JO, Al[ there are R(w), M(w) such that if 11(1 - P)uolb+28(fi) :S R(w) then the solution of the y-equation with initial value y(O) = (1 - P)uo exists in the large and satisfies

Replacing in the x-equation, we find that x(t) remains bounded, and

Ilx(t)IIC2+28(fi) = Ilx(t)llC(o) :S const·ll uolb+28(O), t ~ 0.

Therefore, the null solution is stable in c2+211(n). In the case where f depends explicitly on u, it is not easy to decouple

the system as above, so we use the center manifold theory. Although it is very difficult, in general, to compute explicitly the function 'P of Theorem 3, in our example we have 'P == 0, because F maps P(X) into itself, so that for any r the unique small solution of (18) is the null function.

By Corollary 13, the null solution of (22) is stable (respectively, asymptotically stable, unstable) in c2+211(n) if and only if the null solution of the one dimensional equation

Z'(t) = P F(z(t)), t ~ 0, (26)

is stable (respectively, asymptotically stable, unstable). Setting z(t) = ((t)l, where 1 is the constant function equal to 1 over n, (26) is equivalent to the scalar equation

('(t) = f(((t),O,O), t ~ 0.

Assume that there is kEN such that

akf ajf ak = a k (0, 0,0) =1= 0, -a . (0, 0, 0) for j < k.

U uJ

If either k is odd, or it is even and ak > 0, then the null solution of (22) is unstable; if k is even and Uk < 0, the null solution of (22) is stable in c2+211(n).

202 A. Lunardi

Note that the instability part is trivial, since it is clear that the set of the constant functions is invariant for problem (22), i.e. if the initial datum Uo

is constant, then the solution depends only on t. The same procedure may be performed in the case of the Dirichlet bound

ary condition. In the critical case of stability c = ).1 (where -).1 is the first eigenvalue of ~ with Dirichlet boundary condition), things cannot be simplified as in the Neumann case because now F(P(X)) is not contained in P(X), so that the function c.p is not necessarily null. However, the computations made in [3J in the one dimensional case may be followed also for n > 1, replacing the little Holder spaces by the corresponding Holder spaces, without further modification. We refer to [3J for details.

References

1. P. Acquistapace, B. Terreni, Holder classes with boundary conditions as interpolation spaces, Math. Z. 195 (1987),451-471.

2. C.-M. Brauner, J. Buckmaster, J.W. Dold, C. Schmidt-Laine, On an evolution equation arising in detonation theory, preprint.

3. G. Da Prato, A. Lunardi, Stability, instability and center manifold theorem for fully nonlinear autonomous pambolic equations in Banach space, Arch. Rat. Mech. Anal. 101 (1988), 115-141.

4. A.I<. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Analysis T.M.A. 13 (1989), 1091-1113.

5. D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math. 840, Springer-Verlag, New York (1981).

6. A. Lunardi, Asymptotic exponential stability in quasilinear parabolic equations, Nonlinear Analysis T.M.A. 9 (1985), 563-586.

7. A. Lunardi, Stability in fully nonlinear parabolic equations, preprint.


8. A. Lunardi, Analytic semigroups and optimal regularity in parabolic equations, book in preparation.

9. A. Lunareli, Bounded solutions of linear periodic abstract parabolic equations, Proc. Royal Soc. Edinburgh 110A, 135-159.

10. M. Poitier-Ferry, The linearization principle for the stability of solutions of quasilinear parabolic equations, I, Arch. Rat. Mech. Anal. 77 (1981), 301-320.

11. E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107(1985), 16-66.

12. H.B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1980),299-310.

Dipartimento eli Matematica, Universita eli Cagliari Via Ospedale 72, 09124 Cagliari, Italy

Fractional Integrals and Semigroups

Adam C. McBride

Introduction

In this paper we shall survey some of the connections between

operators of fractional integration and semi groups of operators.

For simplicity we shall restrict attention to operators related to

the Riema.nn-Liouville fractional integral. On the one hand, such

operators provide illustrations of the general theory of semigroups,

particularly fractional power semigroups. On the other hand, it

could be said that the operators have provided the stimulus for

extensions of the general theory.

The paper is divided into four sections as follows.

1. Boundary values of holomorphic semi groups.

2. Fractional powers of certain operators mapping one space

into a different space.

3. Fractional powers of certain operators mapping a space

into itself.

4. a-times integrated semi groups.

1. For Re a > 0 and a suitable function </> we define r:t </>, the

Riemann-Liouville fractional integral of order a of </>, by

205


(1)

206 A. C. McBride

When Q = n, we obtain an operator corresponding to repeated

(n-fold) integration.

First consider the properties of [Ot relative to the spaces LP(O, 1) so that we take 0 < z < 1 in (1). From [9, pp. 664 et seq.] we

know that {lOt : Re Q > O} gives rise to a holomorphic semigroup

of bounded linear operators on LP(O, 1) (for 1 ~ P ~ (0) and the

fird indez law

holds in the sense of operators on LP(O, 1).

On writing Q = JI. + il.l(JI., 1.1 real with J.L > 0), we may consider

the behaviour of F~ as J.L = Re 0: --.. 0+. For 1 E LP(O, 1), the limit

(3)

was well-defined, the limit existing in the LP(O, 1) norm. Further

more, the family of operators {Ii II : 1.1 E R} so defined forms a

strongly continuous group of bounded operators on LP(O, 1). We

can think of this group as giving the boundary valufJ of the orig

inal holomorphic semi group and, as one might expect from (2),

in the sense of operators on LP(O, 1) for 1 0) are now unbounded operators whose

domains are proper subspaces of LP(O, (0). Nevertheless, Fisher

[6] proved that it was still possible to obtain a boundary group

Fractional Integrals 207

{Ii" : II E R} of bounded linear operators on the whole space

LP(O, 00), although the limit process was more elaborate than that

in (3). Equation (4) remains valid in the sense that

whenever

These results have led to more general investigations into bound

ary values of holomorphic semi groups of (possibly) unbounded

linear operators in a Banach space X. We mention in partic

ular the work of Hughes and Kantorovitz [12] who introduced

the concept of a regular umigroup of operatorJ and showed that

such semigroups {T(a) : Rea> O} gave rise to boundary groups

{T( ill) : 11 E R} of bounded linear operators, the boundary groups

being strongly continuous. As indicated in [12], this is one instance

where results for fractional integrals have suggested an extension

of the general theory of semigroups.

2. We have already mentioned that, in the setting of LP(O, 00),

the Riemann-Liouville fractional integral is an unbounded oper

ator. One way to remedy the situation is to introduce weighted

spaces with simple powers as weights. In this section and the

next we shall work within the framework of the Banach spaces

Lp,Jl (1 :11 4> IIp,Jl < oo}

1/ 4> IIp,Jl = {1°C I z-Jl4>(z) IP dz/z};. (5)

It is clear that Lp,Jl is homeomorphic to the usual space LP(O, 00)

under the mapping 4>(z) --+ Z-Jl-l/P4>(Z). (We shall exclude the

208 A. C. McBride

cases p = 1 and p = 00 although many of our results apply in such

spaces also.)

If we take ¢(z) = z~ with ReoX > -1, then

i.e.

(6)

The change in power from oX to oX + a can easily be accommodated

within the structure of our weighted spaces. It can be shown that

[0 i" a continuou" linear mapping from L p ,ll into

Lp,ll+o provided that Re J.' > -1, Re a > O. (7)

Thus each fCx (and [1 in particular) maps from one weighted space

into a different one.

We shall now consider a class of operators, each of which maps

from one Lp ,ll space into a different one. For each such operator T we shall define a general power TO (Re a > 0) and thereby obtain

a fractional power semigroup. The Riemann-Liouville fractional

integral can be recovered as a special case. However, by restricting

attention to smooth functions, we can also treat fractional deriva

tives and powers of "Bessel type" differential operators. We shall

merely outline the theory here. Full details can be found in [21],

[22] and [23], with an edited version in [24, pp. 99-139].

We shall make extensive use of the Mellin transform M defined

formally by

(8)

(For 4> E L",.., the integral exists via mean convergence provided

that 1 of two functions k and 4> is defined by

100 z dt (k * 4»(z) = k( - )4>(t)- (z > 0).

ott (9)

Under appropriate conditions, we obtain

(M(k * 4>))(,) = (Mk)( .. )(M4»( .. ). (10)

For fixed k, we can think of the transform S defined by

(11)

It follows from (10) and (11) that

(M(S</>))( .. ) = (Mk)(s)(M</»( .. ). (12)

It is then possible to obtain the mapping properties of the trans

form S by studying its multiplier Mk and extensive investigations

have been carried out by Rooney [26J, [27]. (A change of variable

relates this to symbols of pseudo-differential operators defined via

the Fourier transform.) Under appropriate conditions on Mk, S

will map Lp,/J into itself and we then call Mk an L p,,.. multiplier.

Now let i be a non-zero complex number. Let

(13)

where S is as in (11). (This means (T</»(z) = z-'1(S4»(z).) Then

from (12) and (13) we get

(M(T</>))(, - i) = (Mk)( .. )(M4»( .. ). (14)

210 A. C. McBride

We now assume that the multiplier Mk has a particular 'fac

torised" form, namely h(& - -y)fh($) for some iunction h. Thus

we finally arrive at operators T which satisfy an equation of the

form

(15)

A simple induction based on (15) gives formally

h($ - n-y) (M(Tn</>))(& - n-y) = h(l) (M</»(I) (n = 1,2, ... )

and this immediately suggests that an operator TO can be defined

by requiring that

a h(s - a-y) (A1(T </>))($ - a-y) = h(,,) (M4»($). (16)

This will work provided that h(" - a-y) / h( $) is an acceptable Lp,p.

multiplier, in which case TO will be a continuous linear mapping

from Lp,p. into Lp,p+o"{ under appropriate conditions. ,\Ve may

feel justified in referring to TO as an ath power of T.

It is worth remarking that a general ath power defined in this

way is not unique. To see this we note that if (15) holds for a

particular function h then it will also hold for each of the functions

hr(r = 1,2,···) where

hr ( s) = exp( 2r7ri.9 / -y )h( s ). (17)

However, as regards (16),

hr(s - a-y) _ (2 .) h(s - a··r) hr(.9) - exp - r7rla h( s)

and, since exp( -2r1ria) =1= 1 in general, we may obtain infinitely

many possibilities for TO.

Suppose now that T satisfies (15) and that we use the same

choice of h throughout (i.e. for the calculation of all powers).

Then it is routine to check the validity of the firlt indez law

(18)

under appropriate conditions. Perhaps more surprisingly it is also

easy to deal with the 6econd indez law

(19)

Indeed, with the same h throughout we calculate (TQ).B by re

placing T,o. and , by TQ,{3 and 0., respectively in (16). This

produces (16) with 0. replaced by o.{3 and (19) is proved formally.

The simplicity contrasts with the difficulties encountered in the

spectral approach which is introduced in the next section.

It is time to illustrate our theory and, as promised, we shall

show first how to recover lQ. All we need to do here (and in

subsequent cases) is to identify, and a suitable h.

We start with T = [I where

and find that (under appropriate conditions, e.g. for E COO(O, (0))

Comparison with (15) gives, = 1 and a suitable choice for h is

1 h(s) = r(1 - sf (20)

212 A. C. McBride

Formula (16) then says that [0 == ([1)0 has to be such that

We can check directly that this gives the operator in (1) so that

the choice of h in (20) is in a sense canonical.

To discuss derivatives we need to use smooth functions. With

D == d~ and 4> E GQ(O, 00), we may integrate by parts to obtain

(M(D4>))(I + 1) = -,(M4»(,).

Comparison with (15) gives I = -1 and a suitable choice for h is

given once again by (20). Then Da has to be such that

Examination of (21) and (22) leads us to conclude formally that

(23)

as one might expect.

It is worth commenting briefly on the validity of our formal

calculations. The multiplier r(1 - 3 )/r(l - 3 + a:) in (21) can

serve as an Lp,1' multiplier if Re a: > 0 (or if a: = 0, in which case

we obtain the identity operator) but not if Re a: < O. When we

restrict attention to smooth functions in a subspace Fp,1' of Lp,p

(see Definition 6.1 in [24], for instance), we discover that (21) can

serve as an Fp,p multiplier for any a and can then proceed to

establish (22) and (23). Again see [24] for further details.

Sometimes we may wish to differentiate or integrate with re

spect to a positive power zm of the variable rather than z itself.


It is possible to define operators I: and D~ which extend the

previous results for m = 1. Thus, in L"p with Rca > 0, (21)

becomes

r(l - .!.) (M(I:!4»)(' - ma) = rc . m ) (M~)(,). (24) J.--+o m

Such generalisations are important in connection with special func

tions.

One way of justifying the last remark is to consider an n-th

order differentiai operator of the general form

(25)

As an example, consider

B - -v-In 211+ln -II v-z Z Z

(26)

The equation BvY = -y is Beuel', equation 0/ order II. Con

sequently the operator (26) is sometimes called c hyper.Beuel n+1

operator. To define TO via (16), we assume that L Cj is real and j=1

that n+1

m =1 L aj - n I> O. j=1

(27)

(If the modulus is zero we are in the situation of the next section.)

On replacing each D in (25) by mzm-l dd ,with m as in (27), a zm

concrete expression can be obtained. See [20] and [24], these pa-

pers having been motivated by Sprinkhuizen-Kuyper [29}. Notice

that for BII in (26) we get m = 2. This explains the appearance of

fractional integrals with respect to z2 in formulae for the Bessel

functions such as Sonine's integral [5, §7.12].

214 A. C. McBride

To summarise our theory, we may say that the formula (15)

leads to a very simple method of defining TO a.nd hence of ob

taining a fractional power semigroup (or even a group). The fact

that T maps between different spaces means that some features

are missing by comparison with what will follow in the next sec

tion. For example, since ),,1 - T is meaningless, we do not have

resolvent operators available. Analyticity with respect to a has to

be ha.ndled at a lower level, with Frechet derivatives unavailable.

Thus we are forced to fix not only 4> E Lp,JI. but also z E (D, 00) a.nd then to investigate (Ta 4»( z) as a function of a. Our final

comment here is that the above results are thirled to the Mellin

transform and the weighted spaces Lp,JI.' In contrast we shall now

turn to the spectral approach for operators mapping a general

Banach space into itself.

3. We now consider operators mapping a space (or a subspace

of it) into the same space. Precisely, let X be a Banach space and

let A be a linear operator whose domain D(.4) and range R(A)

are linear subspaces of X. We shall review the basic method of

defining powers of -A. See [3], [11] and [17].

First recall that, if A is a positive real number and Q IS a

complex number satisfying 0 < Re Q < 1, then

A o - 1 = SlO1rQ -"--d)... . 100 \0-1

1r 0 )"+A (28)

The formal analogue of (28) for operators is

or

(-A)Oz = sin1rQ (XJ )..a-1R()..,A)(-A)zd)" (29) 1r Jo

where R(.\, A) = (>.1 - A)-l and 0 < Re a < 1 as before. For (29)

to exist we certainly need Z E D(A) as well as requiring R(>', A) to

exist for all >. > o. To guarantee convergence of (29) as a Bochner

integral we assume that

(0,00) C p(A), the resolvent set of A

II >'R(>', A) II~ M for all >. > 0

where M is a positive constant.

(30)

We observe that condition (30) is satisfied when A is the in

finitesimal generator of a uniformly bounded Co-semigroup, as a

consequence of the Hille-Yosida Theorem.

In order to define (-A)O for Re a > 0 rather than for the

restricted range 0 < Rea < 1, we first observe that (29) can be

rewri t ten in the form

(-A)Oz = sin1l"a f= >.0-1 [R(>.,A) - >. ](-A)zd>. 1[' 10 1 + >.2

+ sin 1I"2a (-A)z. (31)

However the expression (31) is meaningful for the larger range of

values 0 < Rea < 2. (Basically R(>.,A) - 1:>.21 behaves like

>. -2 as >. -+ 00.) We can therefore use (31) to extend the definition

of (-A)Oz to this larger range for z E D(A). Finally, if a satisfies

n -1 < Re a < n + 1 for a positive integer n, we define (_A)O via

(32)

for suitable z. If for simplicity we assume that D(A) = X and

that A is bounded, then the family {( _A)O : Rea> o} is a

holomorphic semigroup.

216 A. C. McBride

To illustrate this theory we return to our theme of the Riemann

Liouville fractional integral. In considering (6) in the previous

section, we introduced a family of different spaces. Alternatively

we can modify 10 by considering the operator p,o defined by

(I'I'°cf»(z) = z-,,-oIoz"cf>(z)

= z;(:)o foE (z - tt-1t"cf>(t)dt. (33)

In contrast to (6) we find that, for Rea> ° and Re(1] + >.) > -1

In view of (34) it is no surprise that

1'1,0 iJ a contintlotlJ linear mapping from Lp,1l

into itJclf if Re a > 0, Re( 7J + J..L) > -1.

(34)

(35)

The operator (33) is an example of an Erdelyi-Kober operator.

Such operators were studied by Erde1yi and Kober in a series

of papers [4], [15] and [16]. Modifications involving fractional

integrals with respect to ;r:2 rather than z subsequently led to

an elegant method for solving dual integral equations arising in

potential theory. (This is related to the emergence of m = 2 in

connection with the Bessel operator (26).) For further details see

the article by Sneddon in [28] as well as Chapter 7 in [19].

Consider the operator /'1,1 on LP(O, 00) for simplicity. (The

results go through for any Lp,1l space with minor changes.) Take

A = _p,1 in order to define powers of /'1,1. The results which

follow are due to Lamb [18]. As regards (30) we find that, for

>. > 0,

R('x _Pll) = ~I - ~J'l+t,l , ).).2 (36)

provided that Re'7 > -1 (where the first I on the right-hand

side denotes the identity operator!). We remark that (36) can be

checked by showing that both sides have the same Mellin multi

plier in the sense of (12), namely (1] + 1 - J)/(>..'7 + A + 1 - AJ).

We then obtain

II AR(A,-I'I,l) II = II I - !['I+t,l II A

1 1 < 1 + A( ) < 1 + - = 2 for Re A > 0, Re n > -1. - 1 + .,., + 1 - 1 "

This completes the verification of (30). On substituting (36) into

(31) and then using (32) we find that, for Re ex > 0 and Re.,., > -1,

where the operator Ha is defined by

The operator Ha in (37) corresponds to integrating ex times with

respect to log 2:. It is often linked with the name of Hadamard.

In the above example, everything could be calculated explicitly.

However the case of [",2 presents more difficulty and the case of

(I",a)" for general ex and f3 seems hopeless,

4. In this final section we shall consider families of bounded op

erators obtained as fractional integrals of semigroups. Consider a

Co-semigroup {T( t) h 20 of bounded linear operators on a Banach

space X and let A : D(A) --t X be its infinitesimal generator.

Then the abstract Cauchy problem (ACP)

du = Au (t > 0); u(O) = Uo dt

(38)

218 A. C. McBride

has a unique "classical" solution u : [0,00) -+ X for any given

Uo E D(A) and u is given by

u(t) = T(t)uo (t ~ 0).

It is of interest to ask what happens if A does not generate a

Co-semigroup.

Let a be real and positive and let {T(t)h>o be as above. We

define a new family {S(")},~o of bounded linear operators on X by

(39)

the integral converging with respect to the operator norm on

B(X). We shall refer to the family {S(,,)} .. ~o as an a-times inte

grated semigroup.

This concept was introduced in the case a = 1 by Arendt [1] and thereafter the theory has been extended to positive integers a

and finally to all positive values of a. Details can be found in the

papers of Kellermann and Hieber [14], Neubrander [25], Thieme

[30] and Hieber [7], [8]. Basically (39) is an operator version of

(1) and we could express this briefly in the form

(40)

If A denotes the infinitesimal generator of {T(t)}, we may apply

the convolution theorem for the (operator-valued) Laplace trans

form to deduce from (39) that

I.e.

(41)


for all sufficiently large (real)).. It is evident that A not only

generates {T(t)h~o but also generates {S(")},~o in a sense em

bodied in (41). However, it is possible to find operators A which

will "generate" a family {S(,,)} ,~o satisfying (41) without gener

ating a Co-semigroup. We shall use the following definition.

Definition A linear operator A : D(A) -+ X is said to be the

generator of an a-times integrated semi group (for a ~ 0) if

(i) the resolvent set, p(A), of A contains (w, 00) for some

wER

(ii) there exists a mapping 5 : [0,00) -+ X which is strongly

continuous and satisfies

II 5(1) II::; MewlJ (I ~ 0)

(where M is a positive constant) and

R(>.,A) =,\0 100 e->'!5(1)d.s

for)' > max(w,O).

For a = 0, the above definition coincides with the usual

infinitesimal generator in view of the Hille-Yosida Theorem. A

" Hille-Yosida Theorem" can be proved for a > 0 too and the

theory in this case bears some similarities with that for a = O.

For example {5( s)} is uniquely determined by A. There are also

some differences, notably that

for a > 0, D( A) need not be denJe in X. (42)

A useful fact is that

if A generateJ an a-timeJ integrated Jemigroup, then A

generate8 a f3-time8 integrated 8emigroup for all f3 > a.

220 A. C. McBride

This means that we can accommodate more operators as genera

tors by increasing a.

With this in mind let us return to the ACP (38) and sup

pose that A generates an r-times integrated semigroup for some

non-negative integer r. It is well known (see, for instance [2]) that

(38) will have a unique "classical" solution for each Uo E D(Ar+I).

This has been extended to positive non-integral values of a by

Hieber [7].

Theorem Let a ~ 0, t > 0 and assume that A generates an

a-times integrated semigroup {S( s)} ,>0 satisfying

for non-negative constants M,w. Then there exists a unique clas

sical solution of (38) for all 1£0 E D(( _A)Q+t+l).

This result indicates a connection with the theory offrac

tional powers of operators discussed in the previous section. At

this juncture, it is legitimate to ask if there are any important

applications which require the use of a non-integral value of a,

thereby justifying the use of "fractional" integration. To provide

this justification, we mention another result of Hieber.

Consider the ACP (38) for the SchrOdinger equation

~; = i~u (t > 0); 1£(0) = 1£0 (43)

in the Banach space LP(R n) where we take 1 < p < 00 for sim

plicity. Hormander [10] proved that (when defined on its natural

domain) i~ generates a Co-semi group on LP(Rn) if and only if

p = 2. In contrast Hieber [8] has shown that i~ generates an


a-times integrated semigroup on V(R n) if and only if

(44)

(This confirms that a = 0 is only possible when p = 2.) It is of

interest to investigate the space of initial conditions Uo for which

(43) has a unique classical solution u. The "optimal" space turns

out to be the Sobolev space W n+2 ,p. To obtain this when n is

odd it is necessary to use fractional values of a, as use of r-times

integrated semi groups with r a non-negative integer will give a

weaker result which only guarantees existence and uniqueness of

a classical solution for Uo E Wn+3,p.

Conclusion This survey paper has touched on a few of the inter

connections between the Riemann-Liouville fractional integrals,

fractional powers of operators and semigroups of operators. It

may be expected that all three areas will continue to playa role

in the future study of evolution equations and abstract Cauchy

problems.

References

1. W. Arendt. Resolvent positive operators, Proc. London Math. Soc. 54: 321-349 (1987).

2. W. Arendt. Vector-valued Laplace transforms and Cauchy

problems, brad J. Math. 59: 327-352 (1987).

3. A. V. Balakrishnan. Fractional powers of closed opera

tors and semi groups generated by them, Pacific J. Math.

10: 419-437 (1960).

4 A ErdeIyi. On fractional integration and its application

to the theory of Hankel transforms, Quart. J. Math. (Oz

ford) 11: 293-303 (1940).

222 A. C. McBride

5. A. ErdeIyi, W. Magnus, F. Oberhettinger and F. Tri

comi. Higher Trarucendental Fundionl, Vol. 2. McGraw

Hill, New York (1953).

6. M. J. Fisher. Imaginary powers of the indefinite integral,

Amer. J. Math. 93: 317-328 (1971).

7. M. Hieber. Integrated semigroups and differential oper

ators on LP-spaces, Math. Ann. 291: 1-16 (1991).

8. M. Hieber. Laplace transforms and a-times integrated

semigroups, Forum Math. 3: 595-612 (1991).

9. E . Hille and R. S. Phillips. Functional A nalYJiJ and

Semi-groupJ. American Mathematical Society, Providence

(1957).

10. L. Hormander. Estimates for translation invariant oper

ators in LP spaces, Ada Math. 104: 93-139 (1960).

11. H. W. Hovel and U. Westphal. Fractional powers of closed

operators, Studia A/ath. 42: 177-194 (1972).

12. R. J. Hughes and S. Kantorovitz. Boundary values of

holomorphic seruigroups of unbounded operators and sim

ilarity of certain perturbations, J. Funct. Anal. 29: 253-

273 (1978).

13. G. K. Kalisch. On fractional integrals of purely imagi

nary order in LP, Proc. Amer. Math. Soc. 18: 136-139

(1967).

14. H. Kellermann and M. Hieber. Integrated semi groups, J.

Fund. Anal. 84: 160-180 (1989).

15. H. Kober. On fractional integrals and derivatives, Quart.

J. Math. (Oxford) 11: 193-211 (1940).


16. H. Kober and A. ErdeIyi. Some remarks on Hankel trans

forms, Quart. J. Math. (Oxford) 11: 212-221 (1940).

17. H. Komatsu. Fractional powers of operators, Pacific J.

Math. 19: 285-346 (1966).

18. W. Lamb. A spectral approach to an integral equation,

Gla3gow Math. J. 26: 83-89 (1985).

19. A. C. McBride. Fractional Gaiculul and Integral Traru

forou of Generalized Function". Pitman, London (1979).

20. A. C. McBride. Fractional powers of a class of ordinary

differential operators, Proc. London Math. Soc. (3) 45:

519-546 (1982).

21. A. C. McBride. Fractional powers of a class of Mellin

multiplier transforms I, Appl. Anal. 21: 89-127 (1986).


multiplier transforms II, Appl. A.nal. 21: 129-149 (1986).


multiplier transforms III, Appl. Anal. 21: 151-173 (1986).

24. A. C. McBride and G. F. Roach (eds.). Fractional Cal

culu3. Pitman, London (1985).

25. F. Neubrander. Integrated semigroups and their applica

tions to the abstract Cauchy problem, Pacific J. Math.

135: 111-155 (1988).

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and extendability of certain types of operators, Ganad.

J. Math. 25: 1090-1102 (1973).

224 A. C. McBride

28. B. Ross (ed.). Fractional Calculu, and iu Application,.

Springer-Verlag, Berlin (1975).

29. I. G. Sprinkhuizen-Kuyper. A fractional integral opera

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Appl. 152: 416-447 (1990).


University of Strathclyde

Livingstone Tower

26 Richmond Street

Glasgow G1 1XH

SCOTLAND, U.K.

Spectral and asymptotic properties

of strongly continuous semigroups

Rainer Nagel

Abstract: The characterization of the asymptotic behavior of strongly continuous semigroups through spectral properties of its generator is a classical subject. Recently some important progress has been made which will be surveyed in this paper.

Section 1

Some exanlples

The asymptotic behavior of strongly continuous semigroups (T( t) k::o of linear operators on Banach spaces is intimately related to spectral properties. In fact, information on the spectrum 0"( A) of the generator (A, D(A)) is in many situations sufficient in order to describe the qualitative behavior of the operators T( t) as t -+ 00. We are mainly interested in the existence of

P := lim T(t), t-'OV

or, more generally, in the relative compactness of {T( t) : t :::: O} for some appropriate topology. Before starting the systematic discussion we present various examples from different areas of analysis.

First, let A E Mn( ([;) be a complex (n x n)-matrix and consider the semigroup (e t A )t20 generated by A. The exponential stabili ty of this

225


226 R. Nagel

semigroup is characterized through spectral properties in the following classical

1.1 Liapunov Stability Theorem. For a matrix A E Mn( C) and the semigroup (et A )t~O the following assertions are equivalent.

(b) lim edlletAIl = 0 for some € > O. t-oo

(c) The spectral bound seA) := sup{Re" : " E a(A)} satisfies seA) < o.

(d) There exists a positive semidefinite matrix R E Mn( t) such that RA+A*R=-Jd.

For the proof we refer to [H-J] Thm. 2.2.1, but we point out that condition (d) is in fact a "spectral" condition: To see this one looks at the operator LlA defined on the space Mn( t) by

LlA(T) := T A + A*T for T E Mn( t) .

Then it can be shown that (d) is equivalent to (d') 0 is in the resolvent set of LlA and the resolvent R(O, LlA) := -Ll Al maps the cone of positive semidefinite matrices into itself.

Next we consider a strictly positive probability measure J.L on IR and the operator

Mf(x):=ix·f(x), xEIR,

for f E LP(IR,J.L), 1 $ p < 00. Then M with domain D(M) := {J E LP : M f E LP} generates a group of isometries

While the spectrum in each case is a(M) = iIR its fine structure as well as the asymptotic behavior of (T(t»t~O may depend on the measure J.L.

To show this we consider weak operator convergence of T(t) to zero, i.e.,

lim < T(t)f,g >= 0 t-oo

Strongly Continuous Semigroups 227

for all f E LP, 9 E Lq, where q satisfies ~ + ~ = 1. It suffices to verify convergence for all exponential functions

But

where p.(t) := IR eitz dJ.L(x) denotes the Fourier-Stieltjes transform of the measure J.L. But {1. is a continuous function on m. vanishing at infinity whenever J.L is absolutely continuous with respect to the Lebesgue measure. This is the Riemann-Lebesgue lemma (see [Ru2], Thm. 7.5) and will be stated as a convergence result for the above semigroup.

1.2 Riemann-Lebesgue Lemma. If the probability measure J.L on m. is absolutely continuous with respect to Lebesgue measure then the semigroup defined by T(t)f(x) := eitz . f(x) (x Em., t ~ 0, f E LP) satisfies

lim T(t) = 0 t ..... oo

for the weak operator topology on any space LP(m.,J.L), 1 $ p < 00.

Our next example comes from Ergodic Theory and we look at a measure preserving semiflow

¢t : n -- n, t Em., t 2:: 0 ,

on some probability space (n, E,J.L). We refer to [Kr] or [C-F-S] and recall that one of the basic properties of such flows is weak mixing, defined by

llt lim - IJ.L(IP.,(M) n N) - J.L(M)· J.L(N)I ds = 0 t ..... oo t 0

for all measurable sets M, NEE.

This property corresponds to a certain asymptotic behavior of the strongly continuous semigroup of linear operators

T(t)f(x) := f(¢t(x))

228 R. Nagel

for x En and f E U(n,j.L) for any 1 5 p < 00. In perfect harmony with the philosophy of this paper this behavior of the semigroup (T( t) )t~O is characterized by a spectral property of its generator. We collect these standard results from Ergodic Theory (see [C-F-S], Chap. I, §6 & 7) in the so-called

1.3 Weak Mixing Theorem. Let (4)t)t>o be a measure preserving semi flow on a separable probability space (n, f,j.L) and take (T(t»t~O to be the induced linear semigroup on LP(n,j.L), 1 5 p < 00, with generator (A, D(A». Then the following assertions are equivalent.

(a) The flow (4)tk:~o is weakly mixing.

(b) For the semigroup (T( t) )t~O there exists a sequence tj ~ 00 such that

.lim T(tj) = P '-+00

for the weak operator topology with P defined by P f := (In f dj.L) . H, f E U(n,j.L).

(c) The generator (A,D(A» has 0 as the only eigenvalue and the corresponding eigenspace consists of the constant functions.

In the final example we look at a similar situation in Topological Dynamics (see [Au]' Chap.2). Let (4)t)t~O be a continuous semiflow on the compact space X. Among the many properties describing regular or irregular behavior of the flow we recall the one which allows a rather complete description.

The semiflow (4)t)t>o is called equicontinuous (or stable) if {4>t : t ~ O} is an equicontinuous-set of maps from X into X.

Again we look at the induced semigroup of linear operators on the function space C(X) defined by

T(t)f(x) := f(4)t(x))

for x E X and f E C(X). It follows from the Arzela-Ascoli theorem that equicontinuity of the flow corresponds to a compactness property of the induced linear semigroup. We state this in the following


1.4 Theorem. Let (4)t)t>o be a continuous semiflow on the compact space X and denote by (f(t)k~o the induced linear semigroup on C(X). The following properties are equivalent.

(a) The semiflow (4)tk~o is equicontinuous.

(b) The semigroup (T( t) )t~O is relatively compact for the strong operator topology on C( C (X)).

We remark that the above theorem does not give a characterization based on the spectrum of the generator of (T(t))t~o. However, Theorem 1.4 should be compared with the results in Section 4.

After these examples, we now specify our notations and indicate our goals.

Let E be a Banach space and (T(t))t>o be a strongly continuous semigroup of bounded linear operators on E. By (A,D(A)) we denote its genera.tor A with domain D(A). In addition, we use the standard spectral theoretic notation O'(A), PO'(A), p(A), R(>.,A)' for the spectrum, point spectrum, resolvent set and resolvent of A, respectively.

On the space C(E) of all bounded linear operators on E we consider the uniform topology, the strong operator topology and the weak operator topology. It is our aim to characterize - preferably in terms of the spectrum of A - the (relative) compactness of

{T(t):t~O}

in C(E) with respect to uniform, strong and weak operator topology. Of particular importance will be the case that lim T(t) exists in one of

t-oo these topologies.

Section 2

Uniform topology: tbe infinite dimensional Liapunov tbeorem

In order to generalize the Liapunov Theorem 1.1 to semigroups on Banach spaces it is useful to introduce the growth bound

w:= in!{w E lIt : 3M ~ 1 such that IIT(t)1I $ M· ewt "It ~ O}

230 R. Nagel

and the spectral bound

seA) := sup{Re A : A E q(A)}

of a strongly continuous semigroup (T(t»t>o with generator A. It is clear that the conditions (a), (b) and (c) in Liapunov's Theorem are equivalent if

seA) = w,

which is not true in general. We refer to [Nal], A-III for the counterexamples and a detailed discussion of the situation. Here we only state the main result 1,1sing the standard terminology.

2.1 Theorem. Let (T(t))t~o be an eventually norm continuous (e.g., analytic, eventually differentiable or eventually compact) semigroup with generator (A, DCA». Then the following assertions are equivalent.

(a) lim IIT(t)1I = o. t-oo

(b) The growth bound satisfies w < O.

(c) Tile spectral bound satisfies seA) < O.

In Hilbert spaces there is also an analogue of condition l.1.d which can be found in [G-N].

Section 3

Weak operator topology: the Jacobs-Glic1\:sberg-deLeeuw splitting

We now turn our attention to the weak operator topology. The nice feature of this topology is the fact that a semigroup becomes relatively compact with respect to the weak operator topology under quite weak assumptions. These standard facts from Functional Analysis will be stated first.

3.1 Proposition. For a semigroup (T(t»t>o on a Banach space E the following assertions are equivalent. -

(a) {T( t) : t ~ O} is relatively compact in £( E) for the weal.: operator topology.


(b) {T(l)J : t ~ O} is relatively aCE, E')-compact in E Jor every J E E.

(c) {IIT(t)1I : t ~ O} is bounded and {T(t)J : t ~ O} is relatively aCE, E')-compact Jor every J in a dense subset of E.

If the Banach space E is reflexive and {IIT(t)1I : t ~ O} is bounded, then each 01 the above holds. Hence, we have automatic weak operator compactness for bounded semigroups on reflexive Banach spaces. For these an important structure theorem is available yielding a splitting of E and (T(t)) into a 'reversible' and an 'irreversible' part. This theorem follows from the genera.l theory of topologica.l semigroups (see e.g. [Kr] or [Ly]) and we state its version for one-parameter semigroups.

3.2 Theorem. Let (T(t))t>o be a semigroup with generator (A, D(A» on the Banach space E- and assume that (T(t»t~O is relatively compact for the weak operator topology. Then there exists a splitting

into (T(t»-invariant closed subspaces where

(i) the reversible part

Er := lin{J E D(A) : AJ = i).J for some). E lR}

and

(ii) the irreversible part

Eo := {g E E : 0 IS a aCE, E')-accumulation point

of {T(t)g : t ~ O} } .

The action of the semi group (T(t»t>o on the reversible part Er is determined by the eigenvectors J, i.e., AJ = i).f and hence, T( t)f = ei>.t f for t ~ O. Thus (T(t»t>o acts isometrically on these eigenvectors. In fact much more can be ~id.

3.3 Corollary. Let (T(t»t>o be as above and assume E = Er (i.e" (T( t) )t~O has 'discrete spectrum~. Then its strong operator closure

{T(t) : t ~ O} C leE)

232 R. Nagel

is a compact group oj invertible operators.

Such groups are quite well understood. In particular if they consist of positive operators on Banach lattices they can be described explicitly by rotations on solenoidal compact groups (see [Nal]).

The action of (T( t) )t>o on the irreversible part is more difficult to understand. The reason is-that even the existence of a sequence (ti)iElN in lR+ such that

weak- lim T(ti)J = 0 '-+00

does not imply

weak- lim T(t)J = O. t-+oo

Examples for this phenomenon are exhibited by weakly mixing but not strongly mixing flows in Ergodic Theory (see 1.3) and are studied in a more general context in [R-R-S]. Very informative examples can be obtained in the spirit of 1.2 above using interesting facts from Harmonic Analysis. The following is an adaptation from [We] (see also [K-Wl, Chap. I).

3.4 Example. By [Rul], Thm. 5.2.2 there exists a closed, uncountable subset X in r = {z E IC : Izi = I}, called a "Kronecker set", having the property that for some subsequence (ni)iElN in IN one has

.lim zn, = 1 uniformly for z EX. '-+00

Take now a diffuse measure fL supported by X and define the unitary one-parameter group (T(t))tElR. by

T(t)J(z) := zt . J(z)

for f E L2( X, fL) and z EX. Since fL is diffuse the point spectrum of the generator of (T(t))tElR. is empty. This implies, by Theorem 3.2, that weak- lim T(ti)J = 0 for f E L2(X,fL) and some sequence (tdiEi'I in m..

ti--+oo

On the other hand the above property (*) implies that

lim IIT(ni) - Idll = 0, '-+00

and hence, lim T(t) does not exist. t-+oo

Strongly Continuous SeJDigroups

3.5 Corollary. For a relatively weakly compact semigroup (T(t»t~o the following assertions are equivalent.

233

(a) The generator A has no purely imaginary eigenvalues, i.e., pq(A)n iill. = 0.

(b) The semigroup (T(t»t~o has 0 as an accumulation point for the weak operator topology.

Section 4

Strong opera.tor topology: the A-B-L-P theorem and its converse

Convergence and compactness for the strong operator topology is probably the most interesting theoretically and the most useful for applications. We will sketch some of the very recent developments but want to avoid certain technicalities. Therefore, in this section we assume the following.

4.1 Assumptions. Let E be a reflexive Banach space and (T( t) )t>o be a bounded, strongly continuous semigroup on E having genera~r (A, D(A)).

From the boundedness of (T(t))t>o it follows that the spectrum q(A) of A is contained in P E CC : R~)" ~ OJ, i.e., seA) ~ O. Moreover, since E is reflexive, {T(t) : t ~ O} is always relatively compact for the weak operator topology and E splits into a reversible part Er and an irreversible part Eo (see Theorem 3.2). On Er the restricted semigroup becomes relatively compact, even for the strong operator topology, by Corollary 3.3. So, it remains to improve the convergence to o on Eo. Since lim IIT(tj)gll = 0 for some sequence tj - 00 implies

'-00 lim IIT(t)gll = 0 we have the following equivalence. t-oo

4.2 Lemma. Under the above assumptions the semigroup (T(t))t>o is relatively compact for the strong operator topology if and only if the irreversible part Eo is given by

Eo = {g E E : lim IIT(t)gll = O}. t-oo

234 R. Nagel

We call semigroups satisfying this compactness condition (strongly) stable and it is our aim to characterize stability of the semigroup through spectral properties of the generator. We will be guided by the following

4.3 Leitmotif. The convergence "T( t)g -- 0" for 9 E Eo is "better" the "thinner" is the peripheral spectrum,

of the restriction Ao of the generator A to the irreversible part of Eo.

The first step towards our goal and an essential argument in the proofs of the subsequent theorems is based on a classical result of Gelfand in which he extends a well understood phenomena from unitary operators on Hilbert spaces to a class of operators on arbitrary Banach spaces.

4.4 Lemma. such that {IITnll u(T) = {I}.

(see [A-Rj) Let T E £(E) be an invertible operator nEll} is bounded. Then T = Jd if (and only if)

Let now (T(t))tER be a bounded, strongly continuous group on E. Then u(A) is contained in iIR and is non-empty. In fact, for bounded groups a 'weak spectral mapping theorem' holds (see [Na1], A-III, Thm. 7.4). Hence, if u(A) = {a} we obtain u(T(t)) = {I} and the following result. Observe that reflexivity of E is not needed in Lemma 4.4 and Theorem 4.5 .

4.5 Theorem. If (T(t))tElR. is a bounded, strongly continuous group, then T(t) = Id, t E IR, if (and only if) a(A) = {a}.

It is clear that without the boundedness of the group the above equivalence is far from being true (e.g., take T(t) = (~i) with A = (g~) on E = ([;2).

It was Katznelson-Tzafriri, who in 1985, made an important step forward ([K-TJ, see also [PhI]). The semigroup version of their theorem reads as follows (see [Ph2], Cor. 3.4 and [A-Pr], Thm. 3.10).

4.6 Theorem. Let (T(t))t>o be a bounded semigroup whose generator satisfies a(A) n iIR c {O}.-Then

lim IIT(t)(T(s) - Jd)R(,X,A)1I = a t-oo


for all s > 0 and .x > O.

This uniform convergence result implies pointwise convergence of T(t) on all elements of the images of T{s) - Id for all s > O. Since E is reflexive it follows that each T( s) is a mean ergodic operator and E splits into the fixed space of T( s) and the closure of the range of T( s) - I d (see [Kr], Chap. 2, Thm. 1.3). Therefore one obtains the following

4.7 Corollary. Let the Assumptions ../.1 be satisfied. Then the condition

u{A) n iIR C {O}

implies that lim T(t)f exists, in norm, for every fEE. Moreover, t-oo

P:= lim T{t) -# 0 if and only if 0 E Pu{A). t-oo

Shortly after, in 1988, Arendt-Batty [A-B] and independently Lyubich-Phong [L-P] succeeded in obtaining a beautiful and more general spectral condition sufficient for strong convergence and strong compactness.

4.8 A-B-L-P-Theorem. Let (T{t))t~O satisfy Assumption 4.1. If

Pu(A) n iill. = 0 and u(A) n iill. is countable,

then

lim IIT(t)fll = 0 t-oo

for every fEE.

As we have seen above, the semigroup is always stable on its reversible part. Therefore we obtain stability if the spectrum u(Ao) of the restriction Ao to the irreversible part has countable intersection with iill..

4.9 Corollary. Let (T(t))t~O satisfy Assumption 4.1 and denote by Ao the restriction of A to the irreversible part (see Theorem 3.2). If u(Ao) n iill. is countable, then (T(t))t~O is stable.

236 R. Nagel

There are many applications and extensions of the Arendt-BattyLyubich-Phong-Theorem (see [A-Pr] or [B-P]) but we pursue the discussion of its theoretical significance. The first observation to be made concerns the necessity of the above spectral condition for stability.

4.10 Example. Let (T(t»t~O be the shift semigroup, given by

T(t)f(x) := f(x + t), x, t ~ 0,

for f E L2(ill.+,dx). Its generator A has spectrum

q(A) = {A E ([; : Re>. =:; O}, hence q(A) n iill. = iill. is uncountable.

But, the semigroup satisfies

lim IIT(t)fll = 0, t ..... oo

for every f E L2(ill.+, dx), hence is stable.

The search for a condition equivalent to

'q(A) n iill. is countable'

is based on the observation that the spectrum remains unchanged when passing to certain associated operators on new spaces (e.g., on the dual Banach space or on some ultrapower space), but that stability is not preserved in general.

In order to report on the new results in this direction we need the notion of semigroup-ultrapower ELT(t)), for some free ultrafilter U on IN, and corresponding to the strongly continuous semigroup (T(t»t>o on the Banach space E. See [Nal], A-I, 3.6, and [H-R] for precise definitions. There it is shown that (T(t»t>o extends to a strongly continuous semi

group (Tu(t»t~O on ELT(t» wh~se generator Au satisfies O"(Au) = q(A). Thus, if Eu is still a reflexive Banach space (such spaces are ca.lled superreflexive, see [N-R]) and O"(A) n iill. is countable, then it follows from Corollary 4.9 that the ultrapower extension (Tu(t)),;::o remains stable. For such a behavior the following terminology seems to be adequate.

4.11 Definiton. A semigroup (T(t»t>o is called superstable if each ultrapower extension (Tu(t»t;::o is stable; for every ultrafilter U on IN.

Strongly Continuous Semi groups 237

From the above discussion it follows that, on superreilexive Banach spaces, a bounded semigroup with countable peripheral spectrum a(A)n iffi. is superstable. That the converse also holds has recently been shown by [H-RJ, based on previous results by [N-R] for discrete semigroups (Tn)nEi'I'

4.12 Theorem. For a bounded semigroup (T(t))t>o with generator (A, D(A)) on a superreflexive Banach space E the following assertions are equivalent.

(a) The peripheral spectrum a(A) n iffi. is countable.

(b) The semigroup (T( t) )t~O is superslable.

We conclude this survey paper by mentioning some open problems.

Section 5

Open problems

5.1 Weak convergence. No spectral conditions on the generator A characterizing, or related to, the property

"weak- lim T(t)! = 0 for every ! E E" t .... o

seem to be known. On the other hand, for unitary groups on I-Iilbert spaces, we can apply the spectral theorem and are therefore in a situation like in 1.2. See also Chap.l, §7, Theorem 3 in [C-F-S].

5.2 Strong convergence. While

"lim T(t)! = 0, in norm, for every fEE" t-O

is implied by the spectral condition in Theorem 4.8, a spectral characterization of this convergence is still unknown.

5.3 Trivial spectrum implies trivial operator. Such an implication can only hold under precise additional hypotheses as, e.g., in Theorem 4.5. Therefore it is interesing to find other situations allowing similar results. For example, we think of the following:

238 R. Nagel

Let (T( t) )t>o be a strongly continuous semigroup of positive Markov operators on a -Banach space C(X), with X compact. Is it true that u(A) = {OJ implies A = 0 and hence, T(t) = Id for all t ~ O?

For positive groups such results are obtained in [A-G] and [Gr].

[Au] Auslander, J.: "Minimal Flows and their Extensions", North Holland, 1988.

[A-B] Arendt, W., Batty, C.J.K'.: Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837-852.

[A-G] Arendt, W., Greiner, G.: A spectral mapping theorem for oneparameter groups of positive operators in Co(X), Semigroup Forum 30 (1984),297-330.

[A-Pr] Arendt, W., Pruss, J.: Vector-valued Tauberian theorems and asympototic behavior of linear Volterra equations, Siam J. Math. Anal. 23 (1992),412-448.

[A-R] Allan, G.R., Ransford, T.J.: Power- dominated elements in a Banach algebra 94 (1989),63-79.

[B-P] Batty, C.J.K., Phong, V.Q.: Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc. 322 (1990), 805-818.

[C-F-S] Cornfeld, J.P., Fomin, S.V., Sinai, Ya.G.: "Ergodic Theory", Springer-Verlag, 1982.

[Gr] Greiner, G.: A spectral decomposition of strongly continuous groups of positive operators, Quart. J. Math. Oxford 35 (1984),37-47.

[G-N] Groh, U., Neubrander, F.: Stabilitat starkstetiger Operatorhalbgruppen auf C'"- Algebren, Math. Ann. 256 (1981),509-516.

[H-J] Horn, R., Johnson, C.R.: "Topics in Matrix Analysis", Cambridge Univ. Press, 1991.


[H-R] Huang, S., Rabiger, F.: Superstable Co-semigroups on Banach spaces, Preprint (1992).

[Kr] Krengel, U.: "Ergodic Theorems", de Gruyter, 1985.

[K-T] Katznelson, Y., Tzafriri, L.: On power bounded operators, J. Funct. Anal. 68 (1986),313-328.

[K-W1 Kaashock, M.A., West, T.T.: Locally Compact Semialgebras, Mathematics Studies 9, North-Holland 1974.

[Ly 1 Lyubich, Yu.l.: "Introduction to the Theory of Banach Representations of Groups", Birkhauser Verlag, 1988.

[L-P 1 Lyubich, Yu.I., Phong, V.Q.: Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), 37-42.

[Na11 Nagel, R.(ed.): "One-parameter Semi groups of Positive Operators", Lect. Notes Math. 1184, Springer-Verlag, 1986.

[Na21 Nagel, R.: On the linear operator approach to dynamical systems, Conf. Sem. Mat. Univ. Bari.

[N-R 1 Nagel, R., Rabiger, F.: Superstable operators in Banach spaces, Israel J. Math. (to appear).

[Ph11 Phong, V.Q.: A short proof of the Y.Katznelson's and L. TzaJriri's theorem, Proc. Amer. Math. Soc. Preprint (1992).

[Ph21 Phong, V.Q.: Theorems of Katznelson-TzJariri type for semigroups of operators, J. Funct. Anal. 103 (1992),74-84.

[Rul] Rudin, W.: "Fourier Analysis on Groups", Interscience Publ. 1967.

[Ru2] Rudin, W.: "Functional Analysis", McGraw-Hill Book Comp., 1973.

[R-R-S 1 Rosenblatt, J., Ruess, W.M., Sentilles, D.: On the critical part of a weakly almost periodic function, Houston J. Math. 17 (1991), 237-249.

[We] West, T.T.: Weakly compact monothetic semigroups of operators in Banach spaces, Proc. Royal Irish Acad. 67, Sect. A (1968), 27-37.

240

Ra.iner Nagel U niversitat Tiibingen Mathematisches Institut Auf der Morgenstelle 10 7400 Tiibingen Germany Tel.: 0044-7071-293242 e-mail: [email protected]

R. Nagel

Continuation for Quasiholomorphic Semigroups

J. W. Neuberger

Suppose X is a Banach space and T is a strongly continuous semigroup

of members of L(X, X). It has been known for a long time that asymptototic

properties of

IT(t) - II

as t -+ 0 relate closely to smoothness properties of T.

In particular one has that,

lim IT(t) - II = 0 t-o+

if and only if T has a generator which is in L(X, X) (cf [4], p 282). Not so well

known is the fact that if

lim sup IT(t) - II < 2 t-o+

then T is holomorphic ([1]'[2, p 317],[6] [9]). In particular if (2) holds then

AT(t) E L(X, X) for all t > 0

where A is the generator of T defined by

D(A) = {x EX: lim (l/t)(T(t)x - x) exists} t ..... o+

and, if x E D(A),

Ax = lim (l/t)(T(t)x - x). t-o+

Consequently, if

lim sup IT(t) - II < 1, t ..... o+

241

G. R. Goldstein and J. A. Goldstein (eds.). Semigroups of Linear and Nonlinear Operations and Applications. 241-249. © 1993 Kluwer Academic Publishers.

(1)

(2)

(3)

(4)

242 J. W. Neuberger

then (2) holds and, in addition,

T(t)-l E L(X,X)

for some t > o. Hence if ( 4) holds, the generator A of T must be

bounded and so (4) implies (1).

This note concerns implications of the condition

lim inf IT(t) - II < 2. t->O+

We will call semi groups satisfying (5) quasiholomorphic. Examples will show

that (5) does not imply (2) and that (5) itself does not always hold.

Denote by fot(T) (functional of trajectory of T) the set of all functions

9 on [0,00) so that

g(t) = f(T(t)x), t ~ 0.

It is known that ([5], [6]) if T satisfies (5), then fot(T) is quasianalytic in the

sense that no two members of fot(T) agree on an open subset of [0,00). If

fot(T) is quasianalytic, we say that T itself is quasianalytic. Thus the above

references give that every quasiholomorphic semi group is quasianalytic. Observe

that (2) implies that each member of fot(T) is real-analytic on [0,00). The new

result of this note gives that, under (5), members of fot(T) have a continuation

property in the sense that each 9 E jot(T) can be computed from its restriction

to any subinterval of [0,00) (as surely holds under (2)). The ability to do this

extension, assuming (5), is a consequence of recent results in [8).

Some Examples. Suppose that q = nl, n2,... is an increasing sequence of

positive integers and

for all x = (Xl, X2, ... ) E 12 . Then Tq is a strongly continuous semi group on

12 (actually extendable to a group). The following calculates ITq( t) - II :

(5)

(6)

Quasiholomorphic Semigroups 243

Lemma. Suppose (6) holds. Then

ITq(t) -- II = 2· sup I sin(nk t/2)1, t 2 0 (7) k=1,2, ...

Proof of Lemma. Suppose t 2 O. Then

ITq(t) - 112 = sup IIT(t)x - xl1 2

xEi2,llxll=1

00

sup L IXk(exp(ink t) - 112 xEi 2 , IIxll=l k=l

2 sup lexp(ind)-11 2

k=1,2, ...

= 4· sup I sin(nk t/2W k=1,2, ...

so that

ITq(t) - II 22· sup I sin(nk t/2)1· k=1,2, ...

But also,

00

sup L IXk(exp(ink t) - 1)1 2 :S 411x11 2 • sup (sin(nk t/2))2 xEi2 , IIxll=l k=l k=1,2, ...

so that

ITq(t) - II :S 2· sup I sin(nk t/2)1· k=1,2, ...

Therefore

ITq(t) - II = 2· sup I sin(nd/2)1, k=1,2, ...

and the argument is complete.

Example 1. Take w = 1, 2, 3, . .. . Then since

sup I sin(k t/2)1 = 1 k=1,2, ...

it follows that

ITw(t) - II = 2, t 2 O.

244 J. W. Neuberger

Example 2. Take v = 31 , 32 , ... , tm = 271'3-m , m = 1, 2, ... . Then if

mEZ+,

sup I sin(3k • 271' . 3-m )/21 = sine 71' /3) = 31/ 2/2 k=1,2, ...

so that

and hence

Note, however, that

lim sup ITv(t) - II = 2. t-O+

It is clear that Tv is not holomorphic since if it were, then ATv(t) would be

bounded for all t > 0, a contradiction, since A is not bounded and Tv (t)-l E

L(12 , 12 ) for all t ~ O.

The following example shows that fot(T) need not be a quasianalytic col

lection for T a strongly continuous semigroup:

Example 3. Denote by X the Banach space (with sup norm) of all bounded

uniformly continuous functions h from R to C so that h( x) = 0, x s: O.

Define the semigroup T on X by

(T(t)h)(x) = hex - t), hEX, t 2:: 0, x E R.

Now take q E X so that

q(x) = 0, x s: 0, q(x) = x(l- x), 0 s: x s: 1, q(x) = 0, x 2:: 1,

and f E X* so that f(h) = h(I/2), hEX. Define 9 on [0,00) so that

g(t) = f(T(t)q) = h(I/2 - t), t ~ O.

(8)

Quasiholomorphic Semi groups

Then g(O) # 0, but g(t) = 0, t 2 1/2. Since the zero function on [0,00) IS

also in fot(T), it follows that fot(T) is not quasianalytic.

In summary: Not all strongly continuous semigroups are quasianalytic and

there are quasianalytic semigroups which are not holomorphic.

The following is needed in order to express our main result:

Suppose a, b, e E R, b is in the open interval (a, e), f E (0,1), M > 0,

6 E R, 6 # 0, 161 < min(lb - ai, Ie - bl) and 9 is a continuous function whose

domain includes [a, e], We pose:

Problem P( a, b, e, f, M, 6, g) : Find an, ... , am so that

245

It(~)(-l)P-kakl ::;(2-f)P and lapl::;M, p=n, ... ,m, (9) k=O

where

ap =g(a+6p),p=0,1, ... ,n-1, n=[I(b-a)/61]+1, m=[I(e-a)/81l.

We use the following notation: If r = an, ... , am is a solution to problem

P(a, b, e, f, M, 6, g) and q is the set of parameters a, b, e, f, M, 6, g, then

gq,r is the function on [a, e] defined as follows:

(i) gq,r(t) = g(t), t E [a, b],

(ii) gq,r(a+h'p)=ap, p=n, ... ,m,

(iii) gq,r is continuous and linear on each of the intervals [b, a + n6],

[a+p6,a+(p+1)h'], p=n, ... ,m-1, and

(iv) gq,r is constant on [a + m6, e].

Theorem. Suppose T is a strongly continuous nonexpansive semigroup on

the Banach space X, f > 0, and 61, 62 , .•. is a decreasing sequence of positive

numbers converging to zero. Suppose also that

lim sup IT(6k) - II::; 2 - fO. k-+oo

246 J. W. Neuberger

Denote by f a member of X* and by x a member of X so that If I = 1 = Ilxli. Define 9 on [0,00) sothat g(t) = f(T(t)x), t ~ o. Finally, suppose that a,b,c

are three nonnegative numbers so that b is in (a, c) and 0 < 15 < EO. There

is a K E z+ so that if k E Z+, k ~ K, then problem P(a,b,c,E,M,Sk,g)

has a solution qk. Moreover, if for each k E Z+, k ~ K, rk denotes

the sequence {g(a + SkV)}~=o (where w denotes the largest integer such that

(a + SkW) E [a, b]), then

converges uniformly to 9 on some interval [a,dj for some dE (b,c).

The point of the theorem is that the problems:

P(a,b,c,E,M,Sk,g), k ~ K

use only the restriction of 9 to [a, b] in order to construct functions on [a, c]

which agree with 9 on [a, b]. Such a sequence (it is not unique) converges to

9 on a subinterval of [a, c] which is larger than [a, b]. This is our means of

constructively determining 9 outside of [a, b] from the restriction of 9 to [a, b].

Repitition of this process yields a construction of 9 on all of (0,00). This

is our substitute for analytic continuation in the quasiholomorphic case arising

from (5).

Proof of Theorem. Suppose f E X*, x E X, If I = 1 = IIxll and g(t) = f(T(t)x), t ~ O. Note that if n E Z+, S ~ 0, u ~ 0, then

~g(nju,S) = t (~) (-It- kg(u + kS) k=O

= f[(t (~) (-It-kT(S)k)T(u)x] k=O

= J[(T(S) - ItT(u)x]

and so

(10)

Quasiholomorphic Semigroups

By hypothesis,

lim sup IT(ok) - II::; 2 - EO. k-.;.=

Denote by E a member of (O,EO) and denote by Ko a member of Z+ so

that if k > Ko then IT(8k) - II::; 2 - €. Now suppose k> K o, f E X*, x E

X, If I = 1 = Ilxll and g(t) = f(T(t)x), t ~ O. By (10),

Hence

But this is precisely a condition which implies the hypothesis of the main theorem

of [8]. This hypothesis in turn implies the conclusion to the present theorem.

Note that the well-known reduction of the study of a general strongly contin

uous semi group to the study of a nonexpansive semi group yields a similar result

for general strongly continuous linear semigroups.

In [8], [7], a function is said to chaotic at a point x in the interior of D(J)

provided that

lim sup IjJ(S, f, 8) = 2 0-.;.0+

for all subintervals S of D(J) so that x is in the interior of S where

IjJ(S, f, 8) = sup{16 f(n; u, 8)ll/n : [u, u + n8] C S}.

In [8] it is shown that all nonchaotic functions f on an open interval have

the above mentioned extension property. Using this terminology, what we have

demonstrated above is that if T satisfies (5), then all members of fot(T) are

nowhere chaotic on (0,00).

What might be of particular value is a generalization of the above to some

classes of semigroups of nonlinear transformations. This problem was first

247

(11)

248 J. W. Neuberger

mentioned in [6J but no progress has been made in this direction. What gives this

problem particular interest is that some Navier-Stokes systems generate nonlinear

semigroups (cf [3]). It is conjectured that the time of onset to turbulence for

a given trajectory of such a semigroup might occur at the first time at which

a functional of that trajectory becomes chaotic in the above sense. Thus a

condition for nonlinear semi groups which replaces (5) might yield a condition

which precludes turbulence for some trajectories of a Navier-Stokes semigroup.

Quasiholomorphic Semi groups 249

References

[1] A. Beurling. On Analytic extension of Semi groups of Operators. J. Functional

Analysis 6. 387-401 (1970).

[2] A. Beurling. Collected Works Vol. 2. Birkhauser 317-331 (1989).

[3] P. Constantine and C. Foias. Navier-Stokes Equations. Chicago Lec. Math.

(1988).

[4] R. Phillips and E. Hille, Functional Analysis and Semigroups. American

Math. Soc. Colloq. Pubs. XXXI (1957).

[5] D. G. Kendall. Some Recent Developments in the Theory of Denumerable

Markov Processes. Trans Fourth Prague Conference on Information Theory, Sta

tistical Decision Functions, Random Processes. Academia Prague. 11-27 (1967).

[6] J. Neuberger. Quasi-analyticity and Semigroups. Bull. Amer. Math. Soc.

78. 909-922 (1972).

[7] J. Neuberger. Chaos and Higher Order Differences. Proc. Amer. Math.

Soc. 101. 45-50 (1987).

[8] J. Neuberger. Predictability in Absence of Chaos. J. Math. Anal. Appl., to

appear.

[9] J. Neuberger. Beurling's Analyticity Theorem. Math. Intelligencer, to ap

pear.


University of North Texas

Denton,' Texas 76203

USA

Global existence for a reaction-diffusion system with a balance law

Michel PIERRE and Didier SCHMITT

This paper is a contribution to the study of global existence of solutions to reaction-diffusion systems which have the two main following properties :

- first the nonnegativity of the various components of the solution is preserved with time,

- then the total mass of the components is preserved (or more generally non increasing). Many results have already been obtained for these systems

(see [1],[2], [5], [6], [7]). However much needs to be understood yet. For instance, if we restrict ourselves to 2 x 2 systems, global existence is proved only for systems of the above class for which one of the components is a priori uniformly bounded.

Here, we consider a 2 x 2 system for which no a priori knowledge of this sort is easily available, namely

(1) Ut -d1uxx = -c(x)uCl v.8 on (0,00) x (-1,1) (2) Vt - d2 vxx = c(x)uoJ.8 on (0,00) x (-1,1)

(3) u(t,x) =v(t,x) = ° on (0,00) x {-1,1} (4) u(O,x) =uo(x),v(O,x) = vo(x) on (-1,1)

where d1 , d2 are positive constants, ct, (:J > 1

(5) Uo,Vo E LOO(-l,l),uo,vo ~ ° on (-1,1) (6) c E LOO ( -1,1).

It is classical that the system (1) - (4) has local classical nonnegative solutions on some interval (0, T). Moreover, the maximal time of existence Tmax is characterized by the fact that

(7) { (llu(t)lloo + Ilv(t)lloo ~ C, for all t in (0, Tmax)) ~ (Tmax = 00),

251


252 M. Pierre and D. Schmitt

where 11.1100 denotes the LOO-norm on (-1,1). H d1 = d2 , U + v satisfies the linear heat equation so that u

and v are uniformly bounded since, by the maximum principle

Ilu(t) + v(t) 1100 ~ lIuo + voll oo and u, v ~ 0.

By (7), Tmax = 00.

When d1 =f:. d2 , the problem of global existence becomes nontrivial. It is proved in [2] that, if one of the solutions is a priori bounded on any interval (0, T), then so is the other one and global existence holds. It is the case for instance if c ~ ° on (-1,1), in which case, by maximum principle

These systems, with a priori knowledge on one of the components, have been studied in [2], [5], [7].

Here we want to look at the case when the sign of c is not constant on ( -1, 1) so that none of the solutions is a priori known to be bounded. In order to emphasize the effect of a change of sign, we will more precisely assume that

(8) a.e.x E (0,1),c(x) ~ O,a.e.x E (-l,O),c(x) ~ 0.

As first noticed in [3], one can obtain "local" uniform estimates on u and v of the following kind

{for all e in (0,1), for all T ~ Tmax , for all t in (0, T),

(9) for all x in (-l,-e)U(e,l) u(t, x) + vet, x) ~ G(T, f, Uo, vo).

In other word, u and v can only blow up at x = ° for some t*. H so, they have to both blow up at the same time since, by the above recalled results, if one of u or v is bounded, so is the other one.

We do not know yet what happens in general in the situation (8). However, we can prove that, if c vanishes fast enough at x = 0, then global existence holds for (1)-(4). More precisely.

Reaction-Diffusion System

Theorem. Assume (5), (6), (8) hold and

(10) { There exist 'Y > a + (3 - 2 and K > 0, such that for all x in (0,1), c(x) :::; K x'Y.

253

Then, the problem (1)-(4) has a global (classical) solution on (0,00) x (-1,1).

Remarks. Note that condition (10) is unilateral since it only involves the behavior of c for x > 0. Obviously, by symmetry the same conclusion would be obtained under the assumption

(11) { There exist'Y > a + {3 - 2, and K > ° such that for all x in ( -1,0), Ic( x)1 :::; Klxl'Y·

We do not know whether the condition 'Y > a + (3 - 2 is optimal. In particular it would be interesting to know whether global existence holds with c(x) = sign (x). Numerical computations made with d1 = 1, d2 = 0,01, a = {3 = 5 show that the non-linear term u Q v i3 can have a very high peak around x = 0. However, this "peak" seems to disappear after a while.

The proof of the theorem relies on the following lemma together with previous techniques. We set WT = (0, T) x (0,1).

Lemma 1. Assume (5), (6), (8) hold. Then, for all T :::; Tmax and all1 < p < 00, there exists C = C(T,p) such that

(12) { IIxuIILP(wT) + II xv llLP(wT) :::; C(T) (lluo + VollLP(o,l)

+IIu + vllLoo(O,T;£l(O,l)))'

Proof. We argue by duality introducing, for B E Co(WT), B 2 0, the unique solution of the following system :

(13)

(14)

(15)

-('ljJt + d'ljJxx) = B on WT 'ljJ(t,O) = 'ljJ(t, 1) = ° on (O,T)

'ljJ(T, x) = 0 on (0,1).

Recall that 1/J exists, is nonnegative and satisfies for alII < q < 00

(see [4]) (16) l11/JtIIL'l(IoIT) + l11/JzzIIL'l(IoIT) + sup 111/J(t, ·)IIL'l(o,l) ~ CIl8I1L'l(IoIT)·

tE(O,T)

By Sobolev's imbedding theorem, since for all v E w2,Q(0, 1) we have

from (16) we also deduce

(18)

We first use d = d1 in (13). By (1), (8), we have

(19)

We multiply (19) by x1/J which is nonnegative to obtain

(20)

Integration by parts on (0,1) gives

Using boundary conditions (14), (3) and the regularity of u,o/, the integrated part is equal to zero. Therefore, by (20), (21)

We integrate by parts in time to obtain

Reaction-Diffusion System 255

or, going back to (13) (with d = dl ), with p = q',

(24) J.T 1.' xu 8 $ 11"'(0) IlLo(0,1) ·111.10 IILP(0,1) +

2d11luilLoo (O,TjLl(O,l» IltPx IIL1(O,TjLOO (0,1»'

Using (16), (18), the estimate (24) implies by duality that

This gives the first part of the estimate (12). To estimate xv, we use (13) with d = d2 and

(26)

Again, we multiply by xtP, integrate by parts in sr-ace and time and we get

[J.'x",{ -t 1.' xv"', + d2v(x", • .+ 2",.)

+ [ - d2 x tPvx + d2 v(tP + xtPx) - d1XtPUx

+ d1u(tP + xtPx)]~

= - [11 xtPu] ~ + 1T 11 x utPt + d1 u( xtPxx + 2tPx).

Integrated terms are equal to zero. Using the definition of tP, we have

J.T 1.' x v8 $ 1.' x",(O)( Vo + "0)

+ J.T J.' 2",,( d2v + dl u) + xu(.p, + dl "'zz)

~ IltP(O)IIL9(O,1)llvo + uoIILP(O,l)

+ ClltPxll£1(O,TjLOO(O,l»·llv + ullu"'(O,Tj£1(O,l»

+ IlxUIILP(WT)lltPt + d1tPxxIIL9(WT)'

We now apply estimates (16), (18), (25) to deduce by duality

(27) IIxvIILP(wT) ~ C(lluo + VoIILP(O,l) + Ilv + uIILOO(O,T;L1(O,l»)

which, together with (25) yields (12).

Lemma 2. For all T < Tmax

(28) Ilu + vIILOO(O,T;Ll(O,I» ~ Iluo + voIIL1(-l,l)'

Proof. We add (1) and (2), integrate in space on (-1,1) and in time using the boundary conditions (3).

Lemma 3. (see [4]). Assume w is a classical nonnegative solution of

(29) Wt - dwxx ~ aw on QT = (0, T) x (-1,1)

(30) w=Oon(O,T)x{-I,I}

(31) w(O,.) = Wo E L OO( -1,1), Wo ~ °

wbere a is a nonnegative function of Lr(o, T; Lq( -1,1)) witb

(32) { q E [1,00), r E [l~/t' 1!2/t], 1 1 1 - + -2 = - fi,. r q

0< fi, < 1/2

Tben, tbere exists C depending on IlaIILr(o,T;Lq( -1,1», r, q, T, fi,

and I/wol/oo sucb tbat

(33)

Proof of the theorem. We apply Lemma 3 with w = v, d = d2 ,

a(t,x) = l[x>o]KxluO'vP-I. Since c(x) S; ° on (-1,0) and c(x) S; K x' on (0,1), by (2) we do have

(34)

Now, in Lemma 3, we also choose q = 1 and r = 1/(1 - fi, - !) = 1/( 1 - 0") where 0" is chosen in (1/2, 1) so that (see assumption (10))

(35) 8 = ,- (a+f3-1) > -0" >-1.

Reaction-Diffusion System

We compute the Lr(o, T; Ll( -1, l))-norm of a :

J.' [[ a(t,x)dxj' dt ~ K' { dt[[ x6(xu)"(xv)P- 1dx]'

~ K' { dt([ x6/·r [(xu)~(xv)f:; (r(l - 0") = 1).

257

Because of (35), fol x8/0' < 00. We again apply Holder's inequality to obtain

for some p, q < 00.

From (34), (36), Lemmas 1, 2 and 3, we obtain that

(37)

where C(T) depends on T and the data. To finish the proof, we use the usual arguments (see [2], [5])

saying that (26) implies that for all 1 < p < 00

(38)

From (1), (37), (38), we deduce that IluIILOO(QT) ~ C(T) for all T, whence the global existence.


References

[1] A. Haraux, A. Youkana, On a Result of K. Masuda Concerning Reaction-Diff'USion Equations, Tohoku Math. J. 40 : 159-163 (1988).

[2] S. Hollis, R.H. Martin, M. Pierre, Global Existence and Boundedness in Reaction Diffusion Systems, SIAM J. Math. Anal. 18 : 744-761 (1987).

[3] S. Hollis, J. Morgan, Interior Estimates for a Class of Reaction Diffusion Systems from L1 a priori Estimates, JDE 98 : 260-276 (1992).

[4] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Trans. Math. Monographs vol. 23 American Mathematical Society, Providence, R.1. 1968.

[5] R.H. Martin, M. Pierre, Nonlinear Reaction Diffusion Systems, Nonlinear Equations in the Applied Sciences, ed. W.F. Ames and C. Rogers, Academic Press (1991), Notes and Reports in Mathematics in Science and Engineering.

[6] K. Masuda, On the Global Existence and Asymptotic Behavior of Solutions of Reaction Diffusion Equations, Hokkaido Math. J., 12 : 360-370 (1982).

[7] J. Morgan, Global Existence for Semilinear Parabolic Systems, SIAM J. Math. Anal., vol. 20, No.5: 1128-1144 (1989).

Department of Mathematics, B.P. 239 University of Nancy I 54506 - VANDOEUVRE-LES-NANCY France and also supported by URA CNRS 750, Projet NUMATH, INRIA-Lorraine

Convexity of the Growth Bound of

Co-Semigroups of Operators

G. F. Webb

1. Introduction

If etA, t 2:: 0 is a Co-semigroup of bounded linear operators in a

Banach space X with infinitesimal generator A, then the growth

bound of etA, t 2:: 0, defined as a function of A, is wo(A) :=

limt-+oo t log(letAI) (see [4], p. 619). If B is in B(X) (the Ba

nach algebra of bounded linear operators in X), then A + B is the

infinitesimal generator of a Co-semigroup et(A+B), t 2:: 0 in X (see

[7], p. 76). It is thus possible to consider wo(A + B) as a function

of BE B(X) and investigate its properties. The question we inves

tigate here concerns the following property of the growth bound:

If a E (0,1) and B, C E B(X), when is it true that

wo(A + aB + (1- a)C) ~ awo(A + B) + (1- a)wo(A + C). (1.1)

IT X is finite dimensional, then wo(A) = sup{Re'\ : ,\ E u(A)}

(see [U], p. 171). In the finite dimensional case it is known that

if A is an n X n nonnegative irreducible matrix and B and Care

diagonal matrices, then (1.1) does hold (see [8], Corollary 1.1).

That the convexity property (1.1) does not hold in general, even in

the finite dimensional case, may be seen by the following example:

Take X = 1R2, a = !, and

A=[~ !],B=[~ ~],C=[~ ~l. Then wo(A+ !B) = 9.106 > !wo(A+B) + !wo(A+C) = 6.5+2.5.

259


260 G. F. Webb

In [8] Kato proves the convexity property (1.1) under the fol

lowing assumptions: X = C(S), where S is a compact Hausdorf

space, or X = V(S), 1 ~ P < 00, where S is a measure space, A is

the infinitesimal generator of a Co-semigroup of positive operators

in X, and B and C are multiplication operators in X.

Our motivation in investigating the convexity property (1.1)

arises from an age-structured model of a tumor cell population un

dergoing periodic chemotherapy treatment. In Section 2 we show

that the convexity property (1.1) implies that in certain circum

stances shorter periods of treatment are more effective than longer

periods of treatment. In Section 3 we provide some simple sufficient

conditions for (1.1) to hold. The general problem of determining

conditions under which the inequality (1.1) holds, and when it is

strict, remains an interesting open question.

2. A Model of Periodic Chemotherapy

Let n(a, t) be the density with respect to age a of an age-structured

tumor cell population at time t. The total population of tumor cells

at time t is 1000 n(a,t)da. For the untreated tumor cell population

the density n( a, t) satisfies

nt(a, t) + nCl(a, t) = -,8(a)n(a, t), (2.1)

n(O,t) =21000 ,8(a)n(a,t)da, t>O, (2.2)

n(a,O) = 4>(a), a> 0, (2.3)

where ,8(a) is the rate of division of mother cells into two daughter

cells and 4>(a) is the initial age distribution of cells.

The problem (2.1)-(2.3) can be associated with a Co-semigroup

in X := Ll(O, (0). Let,8 E L+(O,oo) and let e~s infCl>o,B(a) =:

Convexity of the Growth Bound 261

(3o > O. Define A : X --+ X by

(A4>)(a) = -4>'(a) - (3(a)4>(a), (2.4)

D(A) = {4> EX: 4>' E X and 4>(0) = 21000 {3(a)4>(a)da}.

It is known that A is the infinitesimal generator of a Co-semigroup

etA,t ~ 0 in X and the formula n(a,t) = (etA4>)(a) provides a

generalized solution to (2-1)-(2.3) (see [11], p 76). It is also known

that etA, t ~ 0 has the following asymptotic behavior which cell

biologists call asynchronous exponential growth (see [11], p. 188):

Let A be the unique real solution to the characteristic equation

Then A = wo(A) and there exists a rank one projection PAin X

and constants M ~ 1 and E > 0 such that

etA = etwo(A)PA + etA (! - PAl, t ~ 0,

letA(! - P A) I ~ M et(Wo (A)-£), t ~ O.

and (2.6)

(2.7)

Now consider a periodic age-specific treatment of the tumor

cell population, that is, a treatment which is periodic in time and

which affects only cells in a certain age range. Such a treatment

corresponds to the addition of a loss term to (2.1). Let a E (0, I),

let Jl. E Loo(O, 00), and for each p > 0 let

p.(a,tjp) = { 0 if a > 0 and 0 $ t $ ap

p.(a) if a > 0 and ap < t < p (2.8)

Let Jl.( a, tj p) be defined for all t ~ 0 by periodicity with period p. The periodic treatment model (with period p) for this off-on type

262 G.P. Webb

periodic treatment is

nt(a,t;p) + nCl(a,t; p) = -(,8(a) +l'(a,t;p))n(a,t;p), (2.9)

n(O,t;p) = 21000 ,8(a)n(a, t; p)da, t > 0, (2.10)

n(a,O;p) = 4> (a) , a> 0. (2.11)

For each p > ° define (B(tip)4>)(a) = -1'(a,t;p)4>(a), 4> E X,

and then (2.8)-(2.10) can be written abstractly as

d dt U(t, S; p)4> = (A + B(t; p))U(t, S; p)4>, t ~ S ~ 0, (2.12)

U(s,s;p)4> = 4> E X,

where U(t, S; p), t ~ s ~ ° is an evolution operator in X. The

question we consider here is whether treatment is more effective

for shorter periods p or longer periods p. We will show that the

convexity property (1.1) implies that treatment in the limiting case

p ---+ ° is more effective than treatment in the limiting case p ---+

00.

Define C : X ---+ X by

(C</>)(a) = -l'(a)</>(a), </> E X (2.13)

and observe that for p > °

H t = np for some integer n, then

and by the Trotter Product Formula (see Lemma 3.1 in Section 3)

we obtain

lim U(t,O;p)</> = et(A+(l-a)C)</>,</> E X. (2.14) p=t/n-O

Thus, for the limiting case P = 0 (which corresponds to continuous

treatment) the value of wo(A + (1 - a)e) determines the growth

(wo(A + (1- a)e) > 0) or extinction (wo(A + (1 - a)e) < 0) of

the tumor.

For P > 0 we assume that the evolution operator U(t,Sjp),t ~

S ~ 0 has the following asymptotic behavior: There exists A(p) E

IR, 1/Jp E X+, 1/J~ E X+, and Ep > 0 such that l11/1p ll = 1, U(p, OJ p)1/1p

= e),(p)P1/1p, and for all tP E X, as t ---+ 00

where u(tjp) = e-),(p)tU(t, OJ p)1/Jp, t ~ 0 (see [3]). The growth or

extinction of the tumor cell population undergoing treatment with

treatment period p is thus determined by the Floquet constant

A(p),

We claim that

lim A(p) = awo(A) + (1- a)wo(A + e). (2.16) p-oo

From (2.6) and (2.7) (which also hold for et(A+C), t ~ 0) we have

that for p > 0

(2.17)

= e(l-O)p(A+C)eopA1/Jp

= [e(l-O)PWo(A+C) P A+C + e(l-O)p(A+C)(I - P A+C)]

[eOPWo(A)PA + eOPA(I - PAl] tPP'

and for some E > 0, as p ---+ 00

264 G. F. Webb

From (2.18) we obtain

lim sup .A(p) ~ o:wo(A) + (1 - o:)wo(A + e). (2.19) p-+oo

We next claim that

(2.20)

From [13], p. 191, PA4> = V(4))e-~47r(a,0)/M, where.A = wo(A),

7r(a, b) = exp[- Ib4 P(u)du], V(4)) = 1000 Iboo p(a)e~(b-4)7r(a, b)4>(b)

dadb, and M = 1000 aP{a)e-~47r(a,0)da. Observe that for 4> E X+

and.A>O

V(4)) = 114>11- .A loo 100 e~(b-4)7r(a, b) 4>(b)dadb

~ 114>11- .A loo 100 e~(b-4)efio(b-4)4>(b)dadb

= .A !Opo"4>I·

For 4> E X+ and .A ~ 0, V(4)) ~ 114>11. Thus, there exists K > 0

such that liP A4>11 ~ KII4>11 for all 4> E X+. Let P A4> = (4), 4>~)4>A

and PA+C4> = (<p,<P~+d<PA+C for all <P E X, where <PA, <PA+C E

X+, II<pAIl = II<pA+clI = 1, <p~, <p~+c E X~, and <p~, <p~+c are

strictlypositiveonX+ (see [6]). Then IIPA+cPAtPpll = (tPp,<P~)(<PA'

<p~+d = IIPAtPplI(<pA,<p~+d ~ K(<PA,<P~+d, which implies (2.20).

Then (2.18) and (2.20) imply

o:wo(A) + (1 - o:)wo{A + e) ~ lim inf .A(p), p-+oo

which together with (2.19) implies (2.16).

We next claim that

wo(A + (1 - o:)e) ~ o:wo(A) + (1 - o:)wo(A + e), o:e(O, 1) (2.21)


The convexity property (2.21) follows directly from Theorem 6.1 in

[8]. We provide a direct proof here, however, in order to establish

necessary and sufficient conditions for the inequality to be strict.

Theorem 2.1. Let {J,J.' e L+, let ess infG>o,8(a) > 0, let A be

defined as in (2.4), and let C be defined as in (2.13). Then (2.21)

holds and the inequality is strict if and only if ",(a) =1= const.

Proof. Let f(a) = ,8(a) exp[- f; ,8 (b) db] , a > O. From (2.5) we

have that

1 = 2 fooo e-wo(A)G f(a)da, (2.22)

1 = 2 fooo e-wo(A+C)G f(a) exp[- foG J.'(b)db]da,

1 = 2 fooo e-wo(A+(l-a)C) f(a) exp[-(l- a) foG J.'(b)db]da.

Let xo(a) = -wo(A + (1 - a)C)a - (1 - a) f; ",(b) db, xl(a) = -wo(A + C)a - f; ",(b) db, and x2(a) = -wo(A)a, a ~ O. Since the

exponential function is convex, for all a ~ 0

exp((l- a)xl(a) + aX2(a)) ~ (1- a) exp(xl(a)) + aexp(x2(a)).

(2.23)

Then, (2.22) and (2.23) imply

2 fooo f(a) exp((l- a)xl(a) + aX2(a))da (2.24)

~ (1- a) 1000 2f(a) exp(xl(a))da + a 1000 2f(a) exp(x2(a))da

= 1 = 2 fooo f(a) exp(xo(a))da.

Thus, (2.24) yields

fooo f(a){exp(xo(a)) -exp((l- a)xl(a) +ax2(a))}da ~ 0, (2.25)

266 G. F. Webb

which yields (2.21). If J.'(a) 'I- const, then xl(a) 'I- x2(a) for some

ao > 0, and the inequality (2.23) is strict for ao. Since f(a) > 0 for

a> 0, (2.25) and (2.21) are also strict. If J.'(a) = const, then it can

be shown easily that wo(A+(I-a)C) = awo(A)+(I-a)wo(A+C), since A and C, and hence etA and et(A+C) commute. I

3. Sufficient Condition for Convexity of the Growth Bound

We suppose that A is the infinitesimal generator of a Co-linear

semigroup in X, B and C are bounded linear operators in X, and

a E (0,1).

Lemma 3.1. For each x in X and uniformly in bounded intervals

of t

lim (e-!;a(A+B)e-!;(l-a)(A+C»)ftx = et(A+aB+(l-a)C)x. (3.1) ft--+oo

Proof. Define a new norm on X by

Then II II and II III are equivalent norms on X and lIetAxll1 ~

etwo(A)lIxlh for x E X, t ~ O. Let Al := A - wo(A)I and

then lIetAl xIII ~ IIxlil for x E X, t ~ O. From [7], Theorem

2.1, p. 495, lIeta(Al +B)xlll ~ etalBllllxll b lIet(l-a)(Al +C)xlll ~

et (l-a)ICh II xII 1, and

for x E X, t ~ O. Define Bl := B -IBhI and C1 := C -IChI. Then lIeta(Al+Bdxlll ~ IIxlib lIet(l-a)(A1 +Cdx ll l ~ IIxllb and


Ilet(Al+aBl+(I-a)Cdxlll ~ IIxlll for x EX, t ~ O. By the Trotter

Product Formula (see [5], p. 53), for each x E X and uniformly in

bounded intervals of t

lim (ef>'a(Al+Bdf>'(I-a)(A1 +cd)nx = et(A1 +aB1 +(I-a)Cd x , n-+oo

which implies (3.1). • Theorem 3.1. Suppose that for every f > 0 there exists

Mf. ~ 1 such for t ~ 0 and n = 1,2, ... ,

l(eta(A+B)/net(l-a)(A+C)/n)nl ~ Mf.et(awo (A+B)+(I-a)wo(A+C)+f).

(3.2)

Then (1.1) holds.

Proof. Let f > O. By Lemma 3.1 and (3.2) there exists Mf. > 0

such that for x E X and t ~ 0

lIet (A+aB+(I-a)C) xII

= lim II (eta (A+B)/net(l-a)(A+C)/n) nxll n-+oo

~ Mf.et(awo (A+B)+(I-a)wo (A+C)+f.) IIxli.

Since wo(A + a.B + (1 - a.)e) = inf{w : there exists Kw such that

for t ~ 0, let (A+a,6+(I-a)C)I ~ Kwewt} (see [4], p. 619), (1.1) must

hold. •

We remark that a sufficient condition for (3.2) to hold is that

et(A+aB) and et (A+(I-a)C) commute for all t ~ O. Another suffi

cient condition for (3.2) to hold is that let(A+B)1 ~ etwo(A+B) and

let(A+C) I :5 etwo (A+C) for all t ~ O.

268 G. F. Webb

Theorem 3.2. Suppose that for x E X, t ~ 0, and s ~ 0,

Then (1.1) holds.

Proof. Let f > 0 and define an equivalent norm II II e on X by

IIxll e = SUPB~O e-B(WO (A+B)+e) lIeB(A+B)xll, x E X. Then lIeB(A+B)xll e ~

eB(wo(A+B)+e)lIxll e for x E X, s ~ O. Further, (3.3) implies that

lIet(A+C)xll e ~ etwo(A+C)llxlle for x E X, t ~ O. Then, for x E

X, t ~ 0, n = 1,2, ... ,

II (eto(A+B)/Ret(l-O)(A+C)/R) Rxll e (3.4)

~ et(owo (A+B)+(l-o)(A+C)+oe) IIxll e'

Since for every f > 0, IIxll ~ IIxll e and there exists Me such that

IIxll e ~ Me II xII , (3.4) implies (3.2), and hence (1.1). I

References

1. L. Cojocaru and Z. Agur, A theoretical analysis of interval drug

dosing for cell-cycle-phase-specific drugs, to appear.

2. B. Dibrov, A. Zhabotinsky, Y. Neyfakh, M. o rlova , and L.

Churikova, Mathematical model of cancer chemotherapy. Pe

riodic schedules of phase-specific cytotoxic-agent administra

tion increasing to selectivity of therapy, Math. Biosci. 73(1985),

1-31.


3. O. Diekmann, H. Heijmans, and H. Thieme, On the stability

01 the cell size distribution II, Hyperbolic Partial Differential

Equations III, Inter. Series in Modern Appl. Math. Computer

Science, Vol. 12., M. Witten, ed., Pergamon Press (1986),

491-512.

4. N. Dunford and J. Schwartz, Linear Operators, Part I: General

Theory, Interscience Publishers, New York, 1957.

5. J. A. Goldstein, Semigroups 01 Linear Operators and Applica

tions, Oxford University Press, New York, 1985.

6. G. Greiner, A typical Perron-Frobenius theorem with applica

tion to an age-dependent population equation, Infinite-Dimensional

Systems, Proceedings, Retzhof 1983, F. Kappel and W. Schap

pacher, eds., Lecture Notes in Mathematics, Vol. 1076, Springer

Verlag, Berlin Heidelberg New York Tokyo, 1984.

7. T. Kato, Perturbation Theory lor Linear Operations, Springer

Verlag, Berlin Heidelberg New York, 1966.

8. T. Kato, Superconvexity 01 the spectral radius, and convexity

01 the spectral bound and the type, Math. Zeit. 180(1982),

265-273.

9. R. Nussbaum, Convexity and log convexity lor the spectral ra

dius, Linear Algebra and Its Applications 73(1986), 59-122.

10. A. Pazy, Semigroups 01 Linear Operators and Applications to

Partial Differential Equations, Springer-Verlag, 1983.

270 G. F. Webb

11. G. Webb, Theory of Nonlinear Age-Dependent Population Dy

namics, Monographs and Textbooks in Pure and Applied Math

ematics Series, Vol. 89, Marcel Dekker, New York and Basel,

1985.

12. G. Webb, An operator-theoretic formulation of asynchronous

exponential growth, Trans. Amer. Math. Soc. 303, No. 2

(1987), 751-763.

13. G. F. Webb, Semigroup methods in populat,~on dynamics: Pro

liferating cell populations, Semigroup Theory and Applica

tions, Lecture Notes in Pure and Applied Math. Series, Vol.

116, Marcel Dekker, New York, 1989, 441-449.


Vanderbilt University

N ashville TN 37240 U.S.A.

Index

A -B-L-P Theorem 233,235,237

A bsor bing set 164

Abstract Cauchy problem 9, 59, 217,220,221

Abstract d' Alembert formula 48

Abstract evolution equation 164

Abstract inhomogeneous Cauchy problem 74

Abstract wave equation 46

Action 28

d'Alembert's formula 49

Age-structured model 260

a-times integrated semigroup 205,218-221

Analytic semigroup 14

Antimaximum principle 115

Approximation Theorem 20, 79-81

Arendt-Batty 235

Asymptotic equipartition of energy 48

Asymptotically 00 - r stable 147

Asynchronous exponential growth 251

Backward difference scheme 62

Balance law 251

Banach (see Picard)

Barenblatt solution 95

271


272

Benilan 63

Bessel's equation 213

Bessel-type differential operator

Bifurcation 99,133

Bound state 40

Boundary conditions

inhomogeneous periodic 88

nonlinear 34,88

Boundary group 206,207

Boundary value 205,206

Boundary value problem 111

Branch of solutions 113

C-regularized

( Co )-semigroup

Cell population

121

10,61

260

Center manifold 195

Center unstable manifold 195

Central Limit Thoerem 25

Chernoff formula 22

Conservation law 65

Conservation of mass 93

Continuation 241

Contraction semigroup 11,61

Convexity 259

Crandall-Liggett Theorem 63

Crandall-Tartar Theorem 94

Critical points 110

Index

208,213

Index 273

D'Alembert's formula 49

Daletskii (see Lie)

Degenerate parabolic problem 83

Degree 115

Difference scheme

backward 62

forward 62

Dirichlet Laplacian 32

Dirichlet problem 109

Discrete spectrum 231

Dissipative 11,63

Dissipative chemical reaction 152

Distribution function 24

Dual integral equations 216

Duality map 68

Duhamel formula 36

Elliptic semilinear problem 100

Ellipticity 18

Energy 16,46

Energy norm 17,60

Entropy condition 66

Equipartition of energy 48

Erdelyi-Kober operators 216

Evolution operator 262,263

Evolution problem 203

Exponentially asymptotically stable 187

Exponentially bounded C-regularized semigroup 121

274 Index

Favard class 78

Fermi pseudopotentials 99

Feynman integral 28

Feynman path formula 27

Finite speed of propagation 61,95

Floquet constant 263

Flow through porous medium 60

Forward difference scheme 62

Forward invariant region 144

Fractional derivative 208

Fractional integral 205,207,208,216,221

Fractional power 205,220,221

Fractional power semigroup 205,214

Gaussian heat kernal 60

Gelfand 234

Generalized domain 78

Generator 10,67,215,217-220,241

C-regularized semi group 121

infini tesimal 10,67,133

Glicksberg 221

Global attractor 162

Global dynamics 162

Global existence 251

Green's function 86

Gronwall-type inequality 77

Index 275

Group

boundary 206,207

unitary 14,67

Growth bound 229,259

Hamilton-Jacobi equation 60

Harvesting 109

Heat equation 59

infinite dimensional 129

Heat semigroup 132

Hilbert-Schmidt operator 38

Hille 62

Hodgkin-Huxley equations 162

Hodgkin-Huxley system 160,163

Holomorphic 241

Holomorphic Co-semigroup 177

Holomorphic semigroup 205-207,215

Hopf equation 66

H-smooth 44

Hyper-Besseloperator 213

Index law 206,211

Infinite dimensional heat equation 129

Infinite speed of propagation 60,95

Infinitesimal generator 10,67,133

00 - r stable 146

Inhomogeneous periodic boundary conditions 88

Integral solution 76

276 Index

Integrated semi group 54

Interpolation spaces 187

Invariant manifold 198

Invariant region

forward 144

Irreversible part 231,233

J acob-Glicksberg-deLeeuw splitting 230

Kato-Kuroda Theorem 45

Katznelson-Tzafriri 234

Kinetic energy 46,60

Kobayashi 63

Kronecker set 232

Laplace operator 204

Laplace transform 218

Laplacian 109

Lebesgue (see Riemann)

Lebesgue decomposition 41

Leeuw, de 230

Leray-Schauder degree 115

Liapunov Stability Theorem 226,228

Lie-Trotter-Daletskii product formula 27

Liouville (see Riemann)

Index

Lipschitz semi norm 63

Lipschitzian 109

Lotka-Volterra system 153

Lyapunov functional 148,149

Lyapunov-Perron approach 186

Lyubich-Phong 235

M-dissipative 11,63

Marcinkiewicz space 92

McArthur (see Rozenwig)

Mean Ergodic Theorem 37

Mellir convolution 209

Mellir transform 208,214

Mild solution 30,76

Multiplication operator 13

Multiplier 209,210,212,217

Multivalued function 65,68

N avier-Stokes equation 31,88

Nikodym (see Radon)

Nodal hypersurface 114

Nonautonomous problem 136

Nonlinear boundary condition 34,88

Nonlinear conservation law 65

Nonlinear eigenvalue problem 99

Nonlinear evolution problem 203

Nonlinear parabolic equations 185

277

278

Nonlinear semigroups 59

Nonlinearity 109

Nonnegative solution 110

Normalized duality map 68

Parabolic equation 185

Partially dissipative system 165

Periodic chemotherapy 260

Peripheral spectrum 234,237

Perron (see Lyapunov)

Perturbation Theorem 17,79,81-83

Phong (see Lyubich)

Picard-Banach Fixed Point Theorem 82

Point interactions 99

Porous medium equation 60,62

Positive semi group 42

Positive solution 112

Positivity preserving semigroup 42,94

Positone 109

Potential energy 46,60

Principle of linearized stability 187

Pseudo-differential operator 209

Quasianalytic collection 244

Quasiholomorphic 241,242

Index

Index

Radially symmetric 112

Radon-Nikodym property 63

Reaction-diffusion system 143,251

Regular semigroup 207

Resolvent identity 71

Reversible part 231,233

Riemann-Lebesgue Lemma 207

Riemann-Lebesgue operator 48

Riemann-Liouville fractional integral 205,207,208,216,221

Rozenwig-McArthur 109

Scattered state 40

Scattering theory 43

Schauder (see Leray)

Schauder Fixed Point Theorem 86

Sectorial 186

Self-adjoint 15

Semigroup 10,241

a-times integrated 205,218-221

analytic 14

C-regularized 121

Co 10,61

contraction 11,61

exponentially bounded C-regularized 121

fractional power 205,214

holomorphic 241

holomorphic Co 177

279

280

Semigroup (continued)

integrated 54

nonexpansive 245

nonlinear 59

positive 42

positivity preserving 42,94

regular 207

strongly continuous 61

type w 71,73

ultrapower 236

Semilinear problem 100

Semipositone 109

Separable function 150

Similarity 13

Singular solutions 100

Singularly pertubed Hodgkin-Huxley equations 163

Smooth bounded region 115

Smoothing 138,139

Solution

Barenblatt 95

branch of 113

integral 76

mild 30

nonnegative 110

positive 112

singular 11 0

stable positive 112

subsolution 109

sub-super 109

unstable positive 112

weak 66

Index

Index

Space

interpolation 187

weak LP 92

weighted 207,208,214

Spatially degenerate parabolic problem 83

Spectral bound 226,230

Spectral Theorem 16

Speed of propagation

finite 61,95

infinite 60,95

Stable 228,236

asymptotically 00 - r 147

exponentially asymptotically 187

00 - r 146

manifold 187

strongly 234

super 236

Stable positive solution 112

Stokes operator 32

Stone's Theorem 16

Strong solution 74

Strongly continuous semi group 61

Strongly elliptic 177

Strongly mixing 232

Strongly stable 234

Sub critical 112

Sub solution 109

Sub-super solutions 109

Superlinear 110

Superreflexive 236

Superstable 236

281

282

Symmetric 15

Symmetry breaking 109

Systems of reaction-diffusion equations 143

Telegraph equation 51

Trajectory 242

Trotter (see Lie)

Trotter product formula 262,267

Trunkation 145

Tzafriri (see Katznelson)

Unitary group 14,67

Unstable 187

Unstable manifold 187

Unstable positive solution 112

Variation of constants formula 186

Variation of parameters formula 29

Volterra (see Lotka)

Wave equation 59

Wave function 27

Weak LP-space 92

Index

Index 283

Weak 227

Weak Mixing Theorem 228

Weak solution 66

Weak Spectral Mapping Theorem 234

Weakly mixing 232

Weighted spaces 187,207,208,214

Well-posed 10,59

Wiener's Theorem 39

Yosida (see Hille)

Yosida approximation 77,90

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