morawetz strauss

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COMMUNICATIONS ON PURE AN D APPLIED MATHEMATICS, VOL. XXV, 1-3 1 (1972)

Decay and Scattering of Solutions of aNonlinear Relativistic Wave Equation*CATHLEEN S. MORAWETZ AND WALTER A. STRAUSS

Courant Institute,N. Y . U. Brown University1. Introduction

Consider a classical solution of(1 ) u t t - Au + m% + gu' = 0,with m > 0, g > 0 in all space time, which has data of compact support a tt = 0. We shall show that this solution decays uniformly in 3-space for largetime like a free solution, that is, like a solution of the corresponding linear equa-tion, g = 0. In fact, there exist two free solutions u+ and u- such that(2) u - * + 0 in the energy norm,as t + fa. his leads naturally to the question of how u- and u+ are re-.lated and thus to the scattering properties of the equation.

During the early nineteen fifties equation (1) was considered by quantumfield theorists as the simplest model of a Lorentz invariant nonlinear interaction.The question of existence in the large for smooth solutions of the Cauchy problemwas resolved by Jorgens [5] in 1961. I n [ I 11, Segal proved that to every freesolution u- there is a solution u of (1) which is asymptotic to u- at t = -a.Thus an operator u- -+u is defined. Similarly there is an operator u+ -+ uat t = +a. Segal conjectured that they can be inverted so as to obtain theScattering operator S u- + + + . He verified this in case the couplingconstant g or the incoming wave u- is sufficiently small (cf. [12]): This resulthas been generalized to other equations by Chadarn [l], [2] and von Wahl [15].Strauss [13] verified the conjecture in case m = 0 by making use of a specialenergy estimate. When m > 0, only the much weaker energy estimate ofMorawetz [6] is available, but we show here that it is sufficient to imply Segal'sconjecture. The case when g = g ( x ) is small a t infinity was treated in [14].

* The work of the first author was carried out at the Courant Instituteof Mathematical Scienccaand supported by Contract No. DA-31-124-ARO-D-365andof the second author by NSF ContractNo. GP-16919 at Brown University. Reproduction in whole or in part is permitted for anypurpose of the United S t a t a Government.

10 972 by John Wiley h Sons, Inc.


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2 CATHLEEN S. M O RAWE T Z AND WALTER A. STRAUSSIn order to describe our results precisely, we define the energy norm

which is constant only for free solutions, and the F-norm

For a solution u = u ( x, t ) of (l), we denote by uo = uo(x , t ) the free solu-tion whose Cauchy data at t = 0 agree with that of u : uo(x , 0) = y ( x ) ,&,/at ( x , 0 ) = + ( x ) . Define .F1as the space of the free solutions such thaty has third derivatives in L1 and second derivatives in Le while + has sec-ond derivatives in L1 and first in Le. Then all elements of Fl have finiteF-norm (cf. Appendix B). Define 9 s the completion of S1 n the F-norm.

THEOREM. If uoE9,hen: (a) u tends to zero uniformly as It1 -+ Q),b)u ha sjn it e F-norm, and (c) there exist fr ee solutions u, and u- satisfiing ( 2 ) .

Whenever (2) is valid for the same u, we shall write Su- = u+ ITHEOREM. S is akjned on all of 9 nd is a homeomorphism of 9 n to 9.THEOREM. If uo ES1 hen u is uniformly O(t-3/2) .The ingredients of the proof are the following: the standard energy estimate

(Lemma l ) , the key estimate from [6] that u4/1xI is integrable over all space-time (Lemma 2), the uniform decay like t-312 of free solutions (Lemma 5), andthe uniform boundedness of solutions of (1) (Lemma 6). These ingredients areused in conjunction with the integral equation obtained by inverting the linearpart of (1) :

t(5) u ( x , t ) = uo(x , t ) - j JR( . - , 1 - )gu3 (y ,7 ) d r d7 *

0

R(x , t ) is the Riemann function of the linear equation, i.e., the free solution withCauchy data R (x , 0 ) = 0, R,(x, 0 ) = 6 ( x ) . In fact, for t > 0 (see [3]),

where J1 is the Bessel function and D = D ( x , ) = 1 for 1x1 < t , D = 0


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SOLUTIONS O F A NO NLINEAR RELATM STIC WAVE EQUATION 3for 1x1 > t . Thus the integral equation (5) expresses u as the sum of the freesolution u o , an integral over the surface of a backward characteristic cone, andan integral over its interior. For Theorems 1 and 2, only the gross behavior ofR is required (Lemmas 7 and 8). For Theorem 3 the oscillatory character of theBessel function is needed.

The proof of Theorem 1 consists of repeatedly jacking up decay estimatesbeginning with the integrability of u/lxl. This leads to

lim inff /u4 dx dt = 0as It1 + 0 for any T, t a rate which is uniform for sufficiently nice solutions(Lemma 3). Next we prove that u is uniformly small in arbitrarily large timeintervals (Lemma 4), after which we prove Theorem 1. If u- is given in 9,a variant of Segals iteration procedure (cf. [l 13) allows the construction of theperturbed solution u for -0 0 < t 5 T for some T (Lemma 11). Startingwith the Cauchy data a t time T, u+ is constructed according to Theorem 1 c) .Lemma 10 asserts that u+ ES. he continuity of S then follows from thepreceding estimates. Furthermore, we prove that S s c in the Frechet sense.Finally, Theorem 3 follows from Lemma 12 which demonstrates the uniformboundedness of the spatial La norm of the second derivatives of u.

The theorem of Goodman [4] and Morawetz [7 ] can now be extended toequation (1) : no nontrivial solution (with data in 9) an vanish in a forwardcone. O n he one hand, for data in 9, his follows from Lemma 12 and theproof of [7]. O n the other hand, if the solution u vanished in a forward cone,then u+ would also vanish there (by domain of dependence), which is impossible.

A more general nonlinear term G(u ) is discussed in Appendix A. If weassume that G(u) vanishes up to the third order as u+ 0 and that

t-T

uG(u) 2 2 + d)G(u)2 , 8 > 0 ,Theorems 1 and 3 remain valid for any uniformly bounded, smooth solution.However, uniform boundedness of general solutions is known only if G ( u )grows at infinity slower than us. Roffman [lo] has given an example of a com-plex-valued solution which does not tend to zero uniformly in case 4rnW + G ( u )is positive but G ( u ) is somewhere negative.For the spatially local decay of solutions of nonlinear equations, see [6] and

The results in this paper were announced in [8].~ 4 1 .

2. Proof of Theorem 1 for Smooth SolutionsI n this section u is a solution of (1) with C initial data u = p, ut = 4

having support in 1x1 k. An integral sign with no domain attached is to betaken over all space. We take g = 1 and t 2 without loss of generality.


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4 CATHLEEN S. MORAWETZ AND WALTER A. STRAUSSLEMMA. Let

(a) E = comtant ,r(b) J (m 2 u4 + 4u4) dss 2$ E ,K

where K is any characteristic cone and & denotes sut$kce measure on K .The proof of Lemma 1 is standard and found by multiplying equation (1)by ut and integrating over suitable domains.LEMMA. r/ u4 dx dt 5 EThis is the starting estimate of decay given in [6] but we repeat its proof.Proof of Lemma 2 : Th e identity

with r = 1x1, is integrated over any slab 0 t 5 T.We obtain

[-I(..+ r-' u)ut dx .~ 1 ~ 't lr-o + l & r - ' ( u 4 + Vul' - d x dt = If0Using Schwarz' inequality on the right and the relation

(u, + r-' u = u," + 2r-' u ( x * V u) + r-' d'= u + div (r-' uox )


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SOLUTIONS OF A NONLINEAR RELATMSTIC WAVE EQUATION 5along with part (a) of Lemma 1, we find the bound 2E for the right-hand side.The left side is positive. Hence,

T

is bounded by 4 E . Let T + 00 and the result follows.LEMMA. Lct E, T,S be positive numbers. Lct

Sl= S+ 4T+ (S+ k + 2T) exp {8TE & - I } .Then here exists a subinterval Z of [S , S,] of length 2 T such that u4 dx dt < E .SSProof: Let

S+OTna , = 1 ( I d d x ) d t .

Then,m mm a ,

n-1 S + k + 2 T n - 0by Lemma 2 and the vanishing of u outside 1x1 s t + k.

Suppose a, 2 for n = 1, 2 , , . . N . Then,N N1 1

E 2 (S+ k + 2Tn)-' 4 E or 2 (a + n)" 5 ~ETE- ' ,where a = (S + k ) ( 2T)- l . But the left-hand sum is greater than log N(a + 1)-1.Consequently, N < ( a + 1) exp {~ETE-1).rom the definition of S , ,2TN< (S+k+2T)exp{8ETE-'} = S l - S - 4 T .Hence there is an interval of length 2 T in [S, ,] for which a , < E.

LEU 4. r f E , T, S are positive numbers, there exists a number S, dependingonly on E, T , S, E , k and the rate of decay of uo , with the following property: thereis a subinttrval Z of [S,,] of length T such that


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6 CATHLEEN S. MORAWETZ AND WALTER A. STRAUSSLemma 4 will be proved using the representation (5). For this purpose we

require three more lemmas,

Lemma 5 is well known and is proved in Appendix B.LEMMA6 . u (x , t ) is uniformly boundcd in all space-tim.Proof: We use the integral representation (5). By Lemma 5 , uo is bounded.

The integral term is the sum of a solid integral 1, and a surface integral 1 s :t

1, =/ 1 k ( p ) u ( y ,. dY Y0 1z--ul


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SOLUTIONS OF A NONLINEAR RELATNISTIC WAVE EQUATION 7If we choose + < q < 2, and combine all terms, we obtain the bound from ( 5 ) :

with y = 3 - 4/q< 1. This implies the boundedness of u.LEMMA. Denote, f o r 0 < , < 1, t,

Thcn

t-t.

where 0 2 4,u(tl) + b( t , ) .-*0 4s t , + 0, und b(t,) and d(tp, ) &fiend ontheenergyof 11.The proof of Lemma 7 is contained in Section 3.Proof of Lemma 4: We break up the integral representation as

u = UO + I (0 , t - T ) + I ( t - T, ) ,where

3

I ( a , 6) = / / R ( x - , t - )U ( y, d y d 7 .a

Choose S so large that, by Lemma 5, luol < +E for t 2 . By Lemma 7(c)with a = 0, together with Lemma 6 , we have

I (0 , t - T)' cT-'~'.


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8 CATHLEEN S. MORAWETZ AND WALTER A. STRAUSSHence we may choose T so that II(0, t - T)I < +E . On the other hand, byLemma 7(b) we have

t-TChoose an interval [ t * - T, *] as in Lemma 3 such that t * 2 T + S, whereE is replaced by e4 ( 3 4 T ) ) 4 . If t * - T < t < t * , then the interval of integra-tion is contained in [ t * - T, t *I. Therefore,Combining terms, we conclude that lul < E for t * - T < < * . This com-pletes the proof.

Proof of Theorem 1(a) for smooth solutions: Let E > 0 satisfjr the inequalityCE + CE'I' < E

and let T > 0 satisfycT-114 + cT-=J2 E .

Th e constants t depend on the energy of u and the behavior of uo but we shallnot make this dependence explicit at this point.By Lemma 4, determine t * such that

IuI < E in [ t * - T, * ] .t**=sup{sI l u l < s in [ t * - T , s ] } .Define

If t * * = 00 , there is nothing to prove. Suppose t * * < 00. Let 0 < 6 < 1and let t * * 5 t t * * + 6. Then by (5) we have, using the notation I ( a , b )of the preceding proof,u = u0 + I ( 0 , t - T ) + I ( t - T, - 1) + Z(t - , t * * ) + I ( t * * , t ) .

By Lemma 5, luola = 0, we have cT-"' since t 2 ** 2 . By Lemmas 6 and 7(c) with

IZ(0, t - T)I CT-'",and, by Lemma 7(c) with a = +,


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SOLUTIONS OF A NONLINEAR RELATIVISTIC WAVE EQUATION 9But t - 1 5 t * + + 8 - 1 < t * * , so that by definition of t * *

IZ(t - T, t - 1)1 s C E @ .Lemma 7(a) impliesFinally for the extreme tip of the cone we use Lemma 7(a) to obtain

IZ(t** , t ) l 5 a ( t - **)(sup lul)where a(t - * ) 6 a ( 8 )+ 0 as 8+ . By Lemma 6, sup lul is bounded.We may choose 8 so small that IZ ( t * * , t ) l < +E. Collecting all these esti-mates, we have

for t* * 2 5 t * * + 8. This is less than E by the choices made at the begin-ning of the proof, contradicting the definition of t * * and completing the prooffor smooth data of compact support.

By Lemmas 1 and 6, the c-norm (3) and the sup-remum norm are finite. For the rest of the F-norm (see (4) ) we useProof of Theorem 1(b):

u = UO + Z(0, - T ) + I( t - T, t ) ,where 0 < T < . To the base we appIy Lemma 7(c) with a = 0:

Integrating with respect to t and interchanging the order of integration, weget

if we choose T so that cT-111 A.To the tip of the cone we apply Lemma 7(a) :


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10 CATHLEEN S. MORAWETZ AND WALTER A. STRAWSIntegrating and interchanging the order we find

By Lemma 4, here exists a to < T such thatllu(7)Ilrn < E for 7 1 1 0

We choose E > 0 so that d Tui(T) < ?T. Squaring the representation of u,taking the supremum in x and integrating in t , we find

Hence, since to < T,

The right side remains finite as T'+ 03. Therefore, so does the left side and theF-norm of u is finite.

Proof of Theorem 1(c): Let uT be the free solution whose Cauchy dataat time t = T coincide with the data of u. We have

whence, as in Lemma l(a),11%' - UTlllr


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SOLUTIONS OF A NONLINEAR RELATIVISTIC WAVE EQUATION 11This completes the proof of Theorem 1 for the case of smooth solutions withcompact Cauchy data.

3. Riemann Function EstimatesProof of Lemma 7: I consists of an integral Is over the surface of a char-

acteristic cone and an integral Ic over the interior of the cone:t-t.

t - 7t - t .

where k(p)= cp-l/*Jl(mp) and pS = (t - ) * - x -yl*. Here and else-where c denotes some constant.

To prove part (a), we pull us out of each space integral, use the bounded-ness of k ( , ~ ) ~nd apply Schwarz's inequality in time.

To prove part (b), consider Ic first. By Holder's inequality applied to thethree factors 1, u and u in space-time,

As for Is ,we integrate the spherical integral by parts:


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12 CATHLEEN S. MORAWETZ AND WALTER A. STRAUSSwhere

and

This proves part (b), making use of Lemmas 1 and 6 .This will be used later. To the surface integral we apply Schwan' inequality:

For the proof of part ( c ) , we shall be more general by replacing u3 by u%.

where the integrals are taken over the range1 X - Y ) = t - T , t - t l ~ T ~ ~ - t f % ,

and dS = paddw, 101 = 1. The first factor on the right siLI: is bounLdd, byLemma l(b). As for the second factor, we first note that

p U l - t - 7 I 5 U l


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SOLUTIONS OF A NONLINEAR RELATIVISTIC WAVE EQUATION 13and introduce the characteristic variables 5 = t - + p, q = i - - .Thus p2 = 7. We also introduce a weight factor:

where 6 = ita By Schwarz' inequality, 1; 5 A B , whereA u dy dT ,

andB = / ) ( q ) k O . ) p u2 va dy dT .

Changing variables, we have

The integral in square brackets is precisely the integral of u2 over a part of acone q = constant. This is bounded, according to Lemma 1. So, since l ( q ) - lis integrable, *

A c( tg) supJ u2 d s ,K Kwhere K runs over all characteristic cones and d$ denotes the usual surfaceelement on K.

In the second factor B , we use the gross asymptotic behavior of k:

Thus

whereD(T)=h(q)q4'" u2 v z dy .

We estimate D(T ) in two parts: q > 6 and q < 6. The part of D(T ) overq < b is less than

const. II4.) II, 114 1 *


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14 CATHLEEN S . MORAWETZ AND WALTER A. STRAUSSFor 9 < 6, we have l ( q ) ~ - ~ J 2 ; l -9Jp . At each point we have the identity

Integration of this identity over the range 0 < q < 6 (that is, over the spheri-cal shell t - - < p < t - ) gives

77-314 u2 v Z d y = 41_8J42 v2 d S - 8jV


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SOLUTIONS OF A NONLINEAR RELATIVISTIC WAVE EQUATION 15DEFINITION.n addition to l lu(7)IJs and llullp defined in the introduction,

and ( u ) defined just above, i t is convenient to define

over all space-time. However we shall also use the same notation when the rangeof t is restricted to a certain time interval. (This is usually clear from the con-text).

The remaining estimates are involved in the scattering properties and fromhere on u, u are any functions.

LEMMA. Let - 3 5 < b 2 + 03. b t u ( x , t ) a n d u ( x , t ) be a pairof arbitrary smooth functions. Let

(a) For any 0 5 a < +,we have

These estimates are also valid f o r functions approximable in the F-norm and (by smooth functions. )-norm

Remark. O n the right-hand sides of these estimates, the norms are takenonly over the range a 5 5 ; on the left-hand sides they are taken over-03 < t < 03. This will be clear from the proof.

Proof Denote by w ( x , t ) the same integral as W u ( x , t ) except that u8 isto be replaced by u 2 v . Then we shall obtain .%?u by setting u = u, andWu - Wv by replacing u2 by (u2 + uu + v 2 ) and u by ( u - ) . Accordingto the definition of the F-norm, llwllp consists of three terms. T o estimate theenergy norm, we apply the energy estimate IIR(t - ) * f I c = l l f l l P , as inLemma 1 , to the function f = u2u. We obtain


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16 CATHLEEN S . MORAWETZ AND WALTER A. STRAUSSwhich implies the desired estimates for this term. To estimate the rest of llwllF(see (4))we write w = w1 + wp , where

w1 = JJ R u ' u d y d - 7 , w 2 = 11 R u 2 u d y d ~ .i i , ] >1 I t-7-1 c

To w1 we apply the Corollary of Lemma 7 with t , = 1 (the slight differencein the time interval of integration does not matter) :

with the same integrand as in the Corollary. Since a < f, ( t - ) -3 /2+3usintegrable. Therefore,

Recall that [ [ u ] ] 5 ( U I ) ~ . This implies Lemma 8(a) for w1 when we setu = u, and Lemma 8(b) for w1 when we replace u2 by uz + uu + uz , u byu - and set a = 0.

Finally, let us estimate wz(x , t ) . We write it as 19 + Z, . As in the proof ofLemma 7, we have

Z s = / /It-rICl p C 1-r

Applying Holder's inequality with exponents 6, 2, 3 or 6, 6, 8 to the inner in-tegral, we get

As for the interior, we have


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SOLUTIONS OF A NONLINEAR RELATIVISTIC WAVE EQUATION 17The integrations are taken over It - 1 < 1 only; the asterisks indicate normover p < It - T I only. But we have the trivial bounds Ilullt + 11ullz cJIuII,.

If we put u = u, we thus have

whence

by definition (8). This completes the proof of part (a).using Sobolev's inequality, \lull: is bounded by llulle so that we getOn the other hand,

whence

This implies part (b).4. The Scattering Operator

Proof of Theorem 1 for weak solutions: Beginning with a smooth free solu-tion u o ( x , t ) with compact initial data and with the corresponding perturbedsolution u ( x , t ) , let us make a note of the uniformities in the earlier proof. InLemmas 1 and 2, the bounds depend only on the energy E; in Lcmma 6 hebound also depends on a bound for uo(x , t ) . As for Lemma 3, let us denote byE , [ u ( t ) ] the energy of u a t time t in the region D of space. Suppose k = k(&, T)is chosen so that

By the standard energy inequality over a cone (cf. Lemma l) , the left side of(9)s at least E l z l , t + k [ u ( t ) ]for any non-negative t ; whence

J 1 ( x , t )4 dx dt < & .Z Iz I> t+ k


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18 CATHLEEN S. MORAWETZ AND WALTER A. STRAUSSThus Lemma 3 remains valid if E is replaced by 212 in the conclusion, providedwe assume (9).

Denote by t ( ~ ) he rate of decay of u, that is, the smallest function sat-isfyingsup I u ( x , t ) i < E for t 2 ( e )Z

(cf. Theorem l(a)). Similarly t o ( & ) is the rate of decay of u o . It is clearfrom the proofs of Lemma 4 and Theorem l(a) for smooth solutions that, for agiven positive E , the lower bound of t ( ~ ) epends only on, say, to(&/lOO),E,and the interval [S,S,] of Lemma 3. Furthermore, the quantities whichdetermine S, are ultimately chosen to depend only on E , E and t,,(&/100).

Keeping these observations in mind, let ~ ~ 5 .et (un0} be a sequenceof free solutions with C2 Cauchy data of compact support which converges touo in 9.This sequence exists because such smooth free solutions are densein S, hile 9,s dense in 9,ee p. 2). By Jorgens theorem, there areglobally defined perturbed solutions u, ,n = 1, 2, . . . , of class C2 whose Cauchydata coincide with the data of un0 at time t = 0. Furthermore, the u, areuniformly bounded in the energy and maximum norms. In particular, u,,, -+u oin the energy norm. So, for any E and T, we may find N = N ( E ,T) uchthat

&sup E[u, (O) - U N ( O ) ] 5-n L N 16TThen k = k ( e , T)may be chosen so that

Therefore, (9) is valid for all u, with n )= N. From the preceding paragraphsit follows that, for any E > 0, there exists t ( ~ ) uch that

where N = N(E ) .However, since the energies of the u, are bounded and

r t

in energy norm for each t, just as was concluded by Jorgens. Therefore,almost everywhere for a subsequence. Thus we may pass to the limit to


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SOLUTIONS O F A NONLINEAR RELATIVISTIC WAVE EQUATION 19

The proofs of parts (b) and (c) of Theorem 1 go through without any change.Remark. In Lemma 10 we shall need the fact that the upper bound for- -~ ~ u , , ( t ) ~ ~ ~t depends only on I~U,~(~)IILt, E and t ( E ) . This may

be seen from the proof of Theorem l(b) . Therefore, there is an upper bound thatis independent of n . Furthermore,

~ u p ~ ~ [ ( u , ( r ) ~ I ,t - 0 as T - t 00.n T

LEMMA.IIun - - l l ~ - - * o .I$ in the notation o th e proof just concluded, un0 --* uo in 9,hen

Proof: We have already shown that llunll F is uniformly bounded. Considerthe difference u, - , written in its integral representation (5). Denote

N ( t ) = IIun(t) - um(tl I I a and N o = l l ~ n d t )- r n o ( f ) I l a *Then (10) implies

where + ( T ) = c( I ~ U , ( T ) (1 + I ~ U , ( T ) [Im)2 is bounded and integrable. Hence,N ( t ) =( N o exp(I+(.)7 ) c N o .

Therefore, u, tends to u in energy norm uniformly for all time. Now, byLemma 8(b ), we have the estimate, using [ [ u ] ] + ( u ) C I I U ~ ) ~

Both terms on the right tend to zero. Hence, u, is a Cauchy sequence in theF-norm.

LEMMA 0. Given uoEF,nd the corresponding u, we have u+ Es,nd


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20 CATHLEEN S. MORAWETZ AND WALTER A. STRAUSSProof u+ has been defined as the unique free solution such that

as t - + +CO. e know that I l u3 ( t ) I ( , ~ ~ ~ ( t ) ~ ~ ~ ~ ~ u ( t ) ~ ~ ~s an integrable func-tion of t . We claim that u,. is given by formula (1 1) . Indeed, the right sidehas finite energy and is asymptotic to u ( t ) in the energy norm as t - +OO.Direct differentation, in the weak sense, shows that it is a free solution, so thatit must coincide with u + ( t ) .To prove the rest of the lemma, we approximate uo by smooth un 0 as before.We denote by u, the corresponding solution of (1) and by unn the free solutionwhose Cauchy data at time t = + n agree with that of u , . Of course theCauchy data of u, at any time are smooth and of compact support. We havethe integral representation

P nUnn(t) = U , o ( t ) - R ( t - ) * U i ( T ) dT ;Jo

whence, if rn < n say,

Consider the F-norm, (4), of the three terms on the right as m, + 00. Thefirst term goes to zero by assumption, the second by Lemmas 8 (b) and 9, whilethe third is bounded by t(IurnII; [ [ u r n ] ] by Lemma 8(a). Th e latter norms aretaken only over the interval [m, O) :[[u,Il2 = sup sup I u,(x, t ) ' +

t > m 2

By the remark preceding Lemma 9, this expression tends to zero as m- 00.Therefore, {unn} is a Cauchy sequence in 9. all the limit w . Passing to thelimit in its integral representation, we see that

w(t)= uo( t ) -J R ( t - ) * ~'(7)T0

m= u ( t ) -l R ( t - ) * us(.) dr = u+( t ) .


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SOLUTIONS OF A NONLINEAR RELATIVISTIC WAVE EQUATION 21LEMMA1 (Cauchy problem at t = - 0 ) . Let u- E F. hen there exists

a finite time T and a unique solution u of (1 ) of Jinite F-norm on - o < t Twhich is asymptotic to u- as t + -a. The fre e solution uT , whose Cauchy data att = T equal that of u, also belongs to 9.urthermore, uT depends continuously o nu- in 9.

Remark. If u3 is replaced by F ( u ) in equation ( I ) , this lemma depends onlyon the property IF(u)I= O(lu1) as u --f 0.

Proof: In analogy to (1 1 ), here is the representationtu - ( t ) = u ( t ) + [ R ( t - ) * u(T) d7

c 53

If u and v are two solutions as in the statement of the lemma, theni~ ( l ) ( t ) = - / - m R ( t - ) * [u s ( . ) - ?(T) ] d r .

Taking the energy norm, we get

Choose S so near -C O that the last integral is less than m . Then u ( t ) = v ( t )for t S,and heqce for t T as well. This proves the uniqueness.

To construct u, we shall solve the integral equation. Since we shall requireuT E9,he construction must exhibit u as the limit of smooth functions. Forthis reason, we first solve the ordinary Cauchy problem with data at a timeS in the distant past, where -C O < S < T. Let us be a free solution withC2 data of compact support. T will be determined later, will depend only on thesize of certain norms of us , and will be independent of S tself. Define

W u ( t )= - R ( t - ) * U(T) d 7 .JSWe solve u = us + 9 u by successive approximations:Each d) s of class Ce because us is.

) norm, (7),modified so that the range of t is -m < 6 T only. In the followingWe shall consider the F-norm, the [[ ]] norm, ( 8 ) ,and the (


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22 CATHLEEN S . MORAWETZ AND WALTER A. STRAWSargument this will be understood. Now define

We shall assume that 8 is small enough to satisfy the conditions specified below.This can be done by choosing T sufficiently near -a. We now claim that

(i) IIJn ) I /F5 ,(ii) [ [u'~)]] s 8 ,

When n = 0, (i) and (ii) are true by definition while ( u s ) -5 k / m by Lemma 1 .\Z7e prove the claim by induction on n. From Lemma 8(a) we have11 u ( n ) - U A g ( I F 5 [ [u(n- l ) ] ] l+a 11 u(n-1) { 1 dn-l)IF + (u(n-1)) )

5 r e l + a k2-6- 1by the induction hypothesis, Choose 8 so small that

c.iei+a2" 5 min (48, 4k) .With this choice of 8 we then have (i) and (ii). To prove (iii), note that u(")is a solution of

(n ) - A u ( d + m2 U(n) + ( U ( n - 1 ) ) 3 = 0Ut tand therefore enjoys the energy inequality (obtained as in Lemma 1 by multi-plying the equation by u:") and integrating) :

where the range of t is taken to be - 0 < t 5 T. he right side is not greaterthan

T2 SUP (I1UP ) 112 II u("- l ) 113s IIU("-')(t) 1 dtt S T --mwhich, by the induction assumption and what we have already proved, is notlarger than 2m-I k202. If 8 5 (+m)ua, this is a t most 3k2, so that m2(u(n))2 4 k Z .?'his proves (iii).

Next, we apply Lemma 8(b) to the differenceu ( n ) - u(n-l) = &(n-l) - u ( n - a ) .


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SOLUTIONS OF A NONLINEAR RELATIVISTIC WAVE EQUATION 23Using (i)-(iii), we obtain

Choosing the coefficient on the right to be less than one, through choice of 8,{dn)} ecomes a Cauchy sequence in the F-norm. Its limit is the solution( I- W ) - l u , . Furthermore, this solution is a Cz function (if ug is) as aconsequence of the estimate

where D is a first or second derivative. If ug has compact support in space, sodoes the solution. Of course, these statements are valid only for -CO < 5 TNext, let {v ,} be a sequence of Cz free solutions of compact support whichtends to u- in 9. Let u be the just-constructed solution of (1) whose dataat time t = - m equals that of u , . The limit of u, as m -+ OC, will be therequired solution. With m > n, consider

We shall take the F-norm in -CO < t 5 T of each term on the right. Thefirst term tends to zero. The second is less than ce11zksi211um- nll by Lemma8(b), the coefficient being less than 1. This term can thus be balanced on theleft side. The final term on the right is estimated as follows. Let E > 0. Since{ u r n } converges in 9,here exists an N = N ( E ) such that, in an obviousmodification of the notation of (8),

for all m. But by Lemma 8(a),

As before, the constant on the right is less than 4 for 8 small. So < 2efor m > n h N(E).Therefore, the F-norm of the last term of (12) is not greaterthan

C P [ [ U , ] ] ! . - . - 5 C P ( z & ) + o .


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24 CATHLEEN 5. MORAWETZ AND WALTER A. STRAUSSIt follows from (12) that {urn} is a Cauchy sequence in the F-norm on ( - 0 0 , r ] .Call the limit u. By passage to the limit we obtain

tu ( t ) = u-(t ) - S _ _ R ( t - ) * u'(T) d~ ;whence ( J u ( t )- - ( t ) l lO+0 as t + -a, s required.urn a t t = T. Then umT is a smooth solution given byLet uT be as stated in the lemma. Let vmT be the free solution equal to

Just as in the above argument, the right side converges in 9-in this case forall time -00 < t < 03. If the limit of vmT in 9 s called w , then w9and

Tw ( t ) = u- ( t ) -J-aR(t - ) #'(T) dT .Since the Cauchy data of w at t = T agree with those of u, w = u T . Thecontinuous dependence in the F-norm is a consequence of the construction:u depends continuously on u- in (- O, TI, and uT on u- in ( - 00, CO).

Proof of Theorem 2 : Given u- E9,he perturbed solution u has beenconstructed in ( - 0 0 , T] n such a way that uT 9 Lemma 11). Thinkingof t = T as the initial time and uT as providing the initial data, Theorem 1asserts that u exists and has finite F-norm for all time. In addition, u+ 9(Lemma 10). Thus S maps 9 nto itself. Each stage of the construction isunique. Since time is reversible, S is one-to-one onto. To prove the continuityof S (and hence of its inverse), take an arbitrary sequence up ' + - in 9.Denote u$" = Sup' , u+ = Su- , and let u'" ) , u be the corresponding solutionsof ( l ) , likewise u g ' , uT . From Lemma 11 , u'"' + in F-norm on ( -00 , T]and u g ' --+ uT in S. y the reasoning of Lemma 9, u'"' + in F-norm forall time. We have shown that

+a@ ( t ) = Ul_"'(t) -1,( t - ) * [U'"'(T)]' dT .By Lemma 8(b) , the right side converges in 9 o u+ (see (11)) which provesthe continuity of S.

Since p x sup u2 .$,a dx2


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SOLUTIONS OF A NONLINEAR RELATIVISTIC WAVE EQUATION 25tends to zero as t --t f o,we have i l ( u ( t )1 --f E (cf. Lemma 1). On the otherhand, we have (2). Therefore, IIu+II: = 2E = Ilu-lls. Tha t is, JlSu-II, = IIu-ll..

COROLLARY.Proof:

S is a Cm-oberator on F in the Frechet sense.To show that S is differentiable, we must show that

E-I[S(u- + E L ) - ( U - ) ]converges in F and that the limit depends continuously and linearly on v- .Let u- and v- belong to F. ecall that

J -m J-mwith slightly abbreviated notation, where u+ = Su- and u denotes the corre-sponding perturbed solution.

First, we assert that there exists a unique solution w of finite F-norm of theintegral equation

1w = V- - R + ( 3 ~ ' ) .$_mIn fact, since l)ullF < 03, a slight modification of Lemma 8(b) shows that theintegral term defines a continuous linear operator in the F-norm. If attentionis restricted to -m < t 5 T with T sufficiently negative, this operator is acontraction. The assertion follows in a well-known manner. I t should be notedthat 11w 11 depends boundedly on (1v- (1 .

Second, let us denote by u") the perturbed solution corresponding to datau- + EV- at t = -a, hen the difference

w ( ' ) = E - l ( u ( e ) - u )satisfies tw ( d = v- - j - , R rc (&)' + d ' ) u + ua ) w ( 0 ) .Subtracting the integral equation for w , we have

t- w = - ] - ,R*[ (u(~) ' + u( ' )u + ua) (w( ' )- w ) + (I((')' + d ' ) u - ue ) w ] .In the proof of Theorem 2, it is shown that as E + 0, I ) u ( ' ) - I J g+0. By asimilar reasoning as in the proof of Lemma 9, it follows that I(w") - 11F+ .Indeed, we have


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26 CATHLEEN S. MORAWETZ AND WALTER A. STRAUSSwhere o(1 ) represents c ( 1 u ( ' ) - ullF 11 u( ') + 2ullF llwllg which tends to zero.Therefore, w(" tends to w in the energy norm. A slight modification of Lemma8(b) implies its convergence in the F-norm.

Third, let us denote u+ = Su- and u y ) = S(u- + E L ) . Then we have

The right side tends in 9 o the limit

w+ = v- - [" * 3 u e w .This formula shows that w+ depends boundedly and linearly on v- and w ,and hence on v- alone. The higher differentiability of S may be proved in asimilar way.

5. Proof of Theorem 3LEMMA2. Let uo E9 ave the Proper0 that initially V u , has finite energy

norm. Then the first den'vatives of u are bounded in the energy norm fo r all time.Proof We must show that the second derivatives of u have uniformly

bounded La-norms in space. Multiply the differential equation by -Aut andintegrate by parts:d Pdt0 = (utt - Au + me u + u3, - b u t ) =- Q ,

where

andQ = 3 ! u ' V u . V u t d x .

This calculation is justified by a limiting process. Since u has finite F-norm,we can estimate


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SOLUTIONS O F A NO NLINEAR RELATM STIC WAVE EQ UATIO N 27We conclude that P ( t ) = P ( 0 ) - Q d7 is uniformly bounded. So Vu, andAu have bounded spatial L2-norms. The same follows for utt = Vu - mzu - u3.

Proof of Theorem 3 : We must show thatM(t )= sup (1 + IIu(s)II,O < a < t

is bounded. Th e fundamental integral equation ( 5 ) ,t~ ( l ) u0( t )+I ( t - ) * u ( T ) ~ T ,0

is again broken up as

where T < at. According to Appendix B, the first term is less than G ( 1 + t ) -3/2.Essentially, the same estimate on R regarded as a free solution is also used inthe next two terms :

where Vz represents all second derivatives. Using Lemma 12, we have

Therefore, on pulling ( t - ) - ~ / S outside the integral, we obtain

and, on pulling I \ u ( T ) llco5 M ( T ) ~ - ~ / ~utside the integral,

To the fourth term we apply Lemma 7(a) :

where kT is a constant depending on T. On one of the three factors we usel l u ( ~ ) l l ~ M ( 7 ) whence

I Z ( t - T,) l t k b SUP 1 ~ 1 ~( t ) .(b -T . t b


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28 CATHLEEN S. MORAWETZ AND WALTER A. STRAUSSNow fix T so large that cT-'I2 < t . Fix E so small that k& < ). Choose

t o so large that lu(x , t ) l < E for all x and for t 2 o . Then the sum of thelast two terms is less than #t-3/eM(t) provided t 2 o + T. Thus,

t / 4M ( t ) t + & M ( t )+ C! M ( T ) ( ~+ T ) ~ ' *~ .0

Since ( 1 + .)-*las integrable, a bound for M ( t ) follows.Appendix A

Consider the equationutt - Au + ma , + C'(u) = 0 ,

where G is a smooth positive function. This equation has a positive definiteconserved energy.

THEOREM.et u(x,t) be a solution with smooth initial data o compact supportwhich is un gon nb bounded, lu (x , t )J5 U. uppose that G satisfies

Then (1 + I t l)31eu(x, ) is unifonnb bounded and there exist q m p t o t i c free solutionsu*(x, t ) which sati gy ( 2 ) .

EXAMPLE.Proof

G(u) = glulp-lu, where g > 0, p 2 .The proof is the same as in the cubic case if us is replaced by G"u)and &P by G(u ) . Assumption (A.l) is used in Lemmas 1 and 2. Lemma 6 hasbeen assumed. Lemmas 7(a) and 7(c) use assumption (A .2) . Lemma 7(b) shouldnow read

and the proof of Lemma 4 correspondingly modified.


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SOLUTIONS O F A NONLINEAR RELATIVISTIC WAVE EQUATION 29To prove Lemma 7(b), we first note that we may assume u 6 4 in (A.3)

for otherwise the extra factors of C(u) can be replaced by C ( U ) . Thus,

by Holder's inequality. The rest of the proof of Theorems 1 and 3 usesassumption (A.2) repeatedly but is otherwise straightforward.

The uniform decay of solutions is also valid for certain p < 3(where C = g1uI"-lr). This follows by adapting the proof of Lemma 7(c) to thiscase.

Remark 2 . The uniform boundedness of solutions is an open problem ifAll smooth solutions of finite energy are uniformly bounded (in space-time)

Remark 1.

p 2 . However, the proof of Lemma 6 carries over to the following case:provided G (u ) 2 for all u ,

IC ' (u ) l 5 c(ua + C ( u ) ) for d l u ,IG'(u)l = O ( G ( U ) - ' + ~ ' ~uI-'+l) as lul --+ co,

fo r some E > 0.Appendix BDecay of Solutions of the Linear Klein-Gordon Equation

THEOREM.f uoE F9 , t h

where c dcpndronlyon m.Here the norm ~ ~ u o ~ ~ F ,s defined through the Cauchy data y, t$ of u,

as the sum of the L1-norms of up to the third derivatives of y and up to thesecond derivatives of t$.


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30 CATHLEEN S. MORAWETZ AND WALTER A. STRAUSSRemarks on the proof. It suffices to prove the estimate for smooth solutions,

which can be done directly from the solution formula, with * denoting spatialconvolution,

aRu0 = R * r$ + - * p ,atand the explicit formula for R, where p, r$ are the Cauchy data at time t = 0.See also Segal [ll], Nelson [9] and von Wahl [15] for other methods. Let usdenote by &r) the integral of r$ over the sphere of center x and radius T.It is easily seen that

Im + I t (181+ 1871+ I + A dr 2 Ilr$llF, *t / 2Writing p2 = t 2 - 2, we have

4 n R * r$ = t-' $ ( t ) + m J 1 (mC1)p-l B(r) drJo.Since Jl(p) = O ( p - l I a ) , the part of the integral over r < t may be estimateddirectly as O(t -3 /2 ) . The rest may be integrated by parts, using J1= J iwhich cancels the term t - l& t ) . The remainder is

The integral may be integrated by parts a second time, by recalling that

Each of the resulting terms may be estimated directly. The term in p is handledsimilarly.

Bibliography[ 11 Chadam, J. M., Asymptotics for IJu = ma u + C(x, t , u, u , t ) , to appear in Ann. Scuola[2] Chadam, J . M., Asymptotic behavior of equations arising in quantum Jield theory, to appear in[3] Courant, R ., and Hilbcrt, D., Methods of Mathematical Physics, Vol. 11, Interscience Publishen,[4] Goodman, R ., O n e-s ad invariant subspaces and domains of uniqueness for hyperbolic squatwnr,[5] Jorgens, K., Das A@agswertproblrm im Crossm fur cine Klassc nichtlinearcr Wcllmglichungcn,[6] Morawetz, C. S., Tim decayfor th e nonlinear Klcin-cwdon equation, Roc. Roy. Soc. A 306, 1968,

Norm.Sup., Pisa.J. Applicable Analysis.1962, p. 695.Proc. A.M.S. 15, 1964, pp. 653-660.Math. Z., 77 , 1961, 1 1 , pp. 295-308.pp. 291-296.


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SOLUTIONS OF A NONLINEAR RELATIVISTIC WAVE EQUATION 31[7] Morawetz, C. S., A uniqueness theorem o r the relatiuisitic wave equation, Comm . Pure AppI. Math.,[8] Moraw etz, C. S. and Strauss, W . A., Asppto t ics of a nonlinear relativistic waue equation, Bull .[9] Nelson, S., O n some solufwns to the Klein-Gordon equation related to an integral of Sonine, Trans. A.

[lo] Roffman, E. H ., Localizcdsolutwns of nonlinear waue equations, Bull . A.M.S. 76, 1970, pp. 70-71.[ I ] Segal, I. E., Quantization and dLspcrsion f o r nonlinear relativistic equations, Proc. Conf. Mathem aticalTheory of Elementary Particles (Dedham, Mass. 1965), M .I. T. Press, pp. 79-108.[I21 Segal, I.E., Dispersion f o r nonlinear relatiuistic equations, IZ,Ann. Sci. &ole Norm. Sup. (4), 1,

1968, pp. 459-497.[I31 Strauss, W . A., Decay and asyn@totics o r nu = F ( u ) ,J. Functional Analysis, 2, 1968, pp. 409-

457.[I41 Strauss, W . A., Decay of solutions of hyperbolic equations with localized nonlinear terms, Atti delConvegno sui Problemi di Evoluzione ( 1 s t . Di Alta Mat., Rome 1970).[IS] von Wahl, W ., ober die klassischc Liirbarkcit &s Cauch~roblemsur nichtlinearc Wellmglcichungmbci klcinen Anfangmrtm und das asymptotische Verhalkn dcr Msungen, Math. Zeit., 114, 1970,

Vol. 16, 1963, pp. 353-362.A.M.S. to appear.M.S., 154, 1971, pp.227-237.

pp. 281-299.Received June, 1971.

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