Communications in Applied and Industrial Mathematics, DOI: 10.1685/2010CAIMXXXISSN XXXX-XXXX, Vol. 1 (2010) XXX (18pp)

Global existence of smooth solutions to a

two-dimensional hyperbolic model of chemotaxis

Cristiana Di Russo1, Roberto Natalini2, Magali Ribot3

1Dipartimento di Matematica, Universita degli Studi Roma Tre,

dirusso@mat.uniroma3.it

2Istituto per le Applicazioni del Calcolo “M.Picone”, CNR,

r.natalini@iac.cnr.it

3Laboratoire “J.A.Dieudonne”, UMR 6621 CNRS, Universite de Nice Sophia Antipolis,

ribot@unice.fr

Abstract

We consider a model of chemotaxis with finite speed of propagation based on Catta-

neo’s law in two space dimensions. For this system we present a global existence theorem

for the smooth solutions to the Cauchy problem. This result is obtained using estimates

of the Green functions of the linearized operators. We illustrate the behavior of solutions

by some numerical simulations.

Keywords: Chemotaxis, hyperbolic-parabolic system, linearized operators,

Green function.

AMS Subject Classification: 92C17, 35B60, 35Q80.

1. Introduction.

Chemotaxis is the phenomenon in which cells, bacteria, or other organ-isms direct their movements according to certain chemicals in their environ-ment. There is a positive chemotaxis when the chemical substance attractan organism and in this case the chemical factor is called chemoattractant,while there is a negative chemotaxis when an organism is driven away fromthe chemical source, and in this case we have a chemorepellent.Chemotaxis is an important means of cellular communication and determi-nates how cells arrange and organize.

It is possible to describe mathematically this biological phenomenon atdifferent levels. The density of the population involved in a chemotacticprocess can be described at macroscopic level through partial differential

Received DD MM 2010, in final form DD MM 2010

Published DD MM 2010

Licensed under the Creative Commons Attribution Noncommercial No Derivatives

C. Di Russo et al

equations of the following form:

(1)

∂tu = ∇ · (µ∇u) −∇ · (χ(u, φ)∇φ) +G(u, φ),

∂tφ = D∆φ+ F (u, φ).

Here u represents the density of the motile living species while φ is theconcentration of the chemical species; the coefficient µ is the motility coef-ficient and the function χ is the chemotactic sensitivity function.We can see that the chemoattractant φ influences the motion of u throughthe transport term ∇ · (χ(u, φ)∇φ), whereas the cell population influencesthe production of chemoattractant through the reaction term F (u, φ).The first model of this class was introduced by Patlak in 1953 in [1]. It wasalso proposed independently by Keller and Segel [2] in 1970, to describethe evolution and pattern formation of cellular amoebae like DictyosteliumDiscoideum. There exist now various models which differ in the choice ofthe chemotactic sensitivity function and of the reaction terms F and G.In addition to the use of these models in the study of biological systems,there is a huge literature about their mathematical properties and, in par-ticular, on the conditions on the model and on the initial data to observeglobal existence of solutions or their blow-up in finite time [3,4].However, one of the major flaws of these diffusive models is the unreal-istic infinite speed of propagation of cells. This type of behavior can beobserved in some large time regimes, but it does not take into account thefine structure of the cell density for short times. So they hardly describesome biological mechanisms, as for instance the phenomenon of angiogen-esis, namely the formation of new blood vessels around and inside of atumoral mass, while a satisfactory description can be given by hyperbolicmodels, as proposed in [5,6].

More recently, with the aim of a more realistic description of these phe-nomena, kinetic models have been proposed to consider the chemotaxis ata mesoscopic level, see [7] and references therein. This approach involveskinetic equations with nonlinear scattering kernels, which are based upon adetailed knowledge of the motion at the cell level. The advantage of thesemodels is that they can give a better description of the bacterial movements,as for instance the characteristic run and tumble movement of EscherichiaColi. From a mathematical point of view, kinetic equations unify the par-abolic and hyperbolic macroscopic models by asymptotic derivations usingeither a diffusion or a hydrodynamical scale as shown in [7,8].Here we are interested in hyperbolic models, which form an intermediateclass of models with finite speed of propagation. They are more accurate

2

DOI: 10.1685/2010CAIMXXX

than the diffusive models but simpler than the kinetic ones. One of themhas been studied by Hillen and Stevens in [9],

(2)

∂tu+ + γ∂xu

+ = −µ+(φ, ∂xφ)u+ + µ−(φ, ∂xφ)u−,

∂tu− − γ∂xu

− = µ+(φ, ∂xφ)u+ − µ−(φ, ∂xφ)u−,

∂tφ−D∂xxφ = −βφ+ α(u+ + u−).

It is a system in one space dimension based on Goldstein-Kac model forcorrelated random walk, and was originally proposed by Segel in [10] todescribe the phenomenon of chemotaxis. At this level, it can be consideredas a kinetic model as well.Here u± are the densities of the right/left moving part of the total popula-tion, φ is the external chemotactic stimulus, µ± are the turning rates whichcontrol the probability of transition from u+ to u− and vice versa, while γis the finite (constant) velocity. It can be noted that the equation for φ isof the same type as the one in system (1), since the sum u+ + u− is equalto the whole population density u.In [9], a first result of local and global existence for weak solutions under theassumption of turning rate’s boundness is proven. Recently Guarguagliniet al. in [11] have proved more general results for this model under weakerhypotheses, by showing a general result of global stability of some constantstates for both the Cauchy problem on the whole real line and the Neu-mann problem on a bounded interval. These results have been obtainedusing the linearized operators and the accurate analysis of their nonlinearperturbations.

In this work we present a first analytical study on the existence ofsmooth solutions for the following hyperbolic-parabolic system:

(3)

∂tu+ ∇ · v = 0,

∂tv + γ2∇u = −v + u∇φ,

∂tφ = D∆φ+ αu− βφ,

where u(x, t) : R2 × R

+ → R, v(x, t) = (v1(x, t), v2(x, t)) : R2 × R

+ → R2

is the flux of u and φ(x, t) : R2 × R

+ → R.This system is the simplest instance of the models proposed by Hillen andDolak in [12] where they considered chemosensitive movements of two dif-ferent species of bacteria, based on Cattaneo’s law of heat propagation with

3

C. Di Russo et al

finite speed. The Cattaneo’s law was introduced by C. Cattaneo in 1948 [13],as a modification of the standard Fourier’s law for the heat conduction, togive a finite speed description of heat propagation.There are two ways to derive hydrodynamical models. The first one is touse Continuum Mechanics to derive balance laws and to use specific stateequations to close the system, see for instance [5], [6].In this paper we follow a different approach where, starting from a kineticequation, it is possible to obtain the hydrodynamical model by a momentclosure technique which minimizes a given functional (in this case the L2-norm of the solution).More precisely we follow the approach proposed in [12], where bacteriamove according to the following rules: an individual cell moves with nearlyconstant speed in a certain direction. It suddenly stops, rotates and choosesa new direction at random. The stopping times can be modeled by an ex-ponential distribution with rate µ . The rotation-times are small comparedto the periods of movement. The distribution of new chosen directions isdenoted by T (v, v′), where v, v′ ∈ V are the incoming and the outgoing ve-locity, respectively. The set V ∈ R

n is the set of possible velocities and theyhave V = sSn−1, where s is the cell speed. It has been shown in [14] thatthis process leads in an appropriate limit to a linear transport equation

(4) pt + v · ∇p = −µp+ µ

∫

T (v, v′)p(v′)dv′,

where p(t, x, v) denotes the particle density at time t ≥ 0 and at spatialposition x ∈ Ω of particles moving with velocity v ∈ V .Starting from (4) and the moment closure procedure, with appropriatechoices of the turning rate and the turning kernel T we can obtain sys-tem (3). More details about this approach can be found in [15] and [12].In [12], numerical schemes and simulations of the biological phenomenon ofpattern formation are presented and discussed.System (3) is also a two-dimensional version of system (2). Indeed, usingas variables the total population density u = u+ + u− and the mean fluxv = γ(u+ − u−), we find a one-dimensional version of (3). Anyway theequations (3) can be seen as a linearized system in the differential part ofthe model proposed in [5].

In this paper we try to give a more rigorous framework to this approach.In Section 2 we present a result of global existence of smooth solutions tothe Cauchy problem in R

2 for small initial data. This result is obtained byusing a detailed analysis of the linearized problem and Duhamel’s formula.In Section 3 we present a preliminary numerical approximation by using

4

DOI: 10.1685/2010CAIMXXX

finite differences methods. Some simulations with different choices of ini-tial data and in a special case with a logistic chemotactic sensitivity arepresented and discussed.

2. The Cauchy problem

In this section we investigate the existence and the behavior for largetimes of global smooth solutions to the Cauchy problem on the whole realplane for system (3), when initial data are sufficiently small in some suitablenorms. For the sake of simplicity we deal with system with D = 1.We complement the system (3) with the initial conditions

u(x, 0) = u0(x), v(x, 0) = v0(x), φ(x, 0) = φ0(x)

where u0 ∈ L1(R2) ∩Hs(R2), v0 ∈ (L1(R2) ∩Hs(R2))2 and φ0 ∈ L1(R2) ∩Hs+1(R2).Let us set ‖w‖Lp = ‖w‖Lp(R2×(0,t)), ‖w(t)‖Lp = ‖w(t)‖Lp(R2) and denote by

Dkx any space derivative Dr

x such that |r| = k.

Theorem 2.1. There exists an ε0 > 0 such that, if

‖u0‖Hs , ‖u0‖L1 , ‖v0‖Hs , ‖v0‖L1 , ‖φ0‖Hs+1 , ‖φ0‖W 1,∞ ≤ ε0

the Cauchy problem (3) has a unique global solution

u ∈ C(R+;Hs(R2)), v ∈ C(R+;Hs(R2)2), φ ∈ C(R+;Hs+1(R2)) for s ≥ 2.

(5)

Moreover for the solutions (u, φ) the following decay rates are satisfied(6)

‖u(t)‖L∞ ∼ t−1 ‖Dkxu(t)‖L2 ∼ t−

k+1

2 for k = 0, 1, 2;

‖φ(t)‖L∞ ∼ t−1

2 ‖D1xφ(t)‖L∞ ∼ t−1 ‖Dk

xφ(t)‖L2 ∼ t−k+1

2 for k = 0, 1, 2.

2.1. Local existence

We prove the local existence of solutions of (3) with a fixed pointmethod. Let us consider the dissipative hyperbolic part of the system (3),

(7)

∂tu+ ∇ · v = 0,

∂tv + γ2∇u = −v.

By Fourier transform methods it is possible to analyze in detail the behaviorof its Green function, see for instance [16] for a complete result in the caseof general dissipative hyperbolic systems.

5

C. Di Russo et al

Let us introduce the vector variable W = (u, v1, v2).Thanks to the Green functions and the Duhamel principle we know that itis possible to write the solutions to (3) as

W(x, t) = (Γh(t) ∗W0)(x) +

∫ t

0Γh(t− s) ∗ F (W, φ)(s)ds,(8)

φ(x, t) = (e−βtΓp(t) ∗ φ0)(x) +

∫ t

0e−β(t−s)Γp(t− s) ∗ g(W)(s)ds(9)

where Γh is the Green function of the dissipative hyperbolic problem (7)and Γp is the Green function of the usual heat equation, g(W) = αu andF (W, φ) = [0, u∂x1φ, u∂x2φ]t .It is possible to prove the local existence of solution with a fixed pointmethod. As a matter of fact we can define an invariant set and build on ita contractive map; thanks to the contraction theorem there is a fixed pointthat will be a local (in time) solution to system (3). More details about thisproof can be found in [17].

2.2. Continuation principle

A major tool to prove global existence of solutions is given by the fol-lowing Continuation principle.

Proposition 2.1. We consider the Banach space

B = C[(0, t);Hs] × C[(0, t);Hs] × C[(0, t);Hs] × C[(0, t);Hs+1]

and let (W0, φ0) ∈ B. If there is an a priori bound on the norms ‖W(t)‖Hs

and ‖φ(t)‖Hs+1 on the interval 0 ≤ t ≤ T ≤ ∞, then the local solution(W, φ) ∈ B can be continued for all t s.t. 0 ≤ t ≤ T .

Proof. Let (u, v, φ) be a given local smooth solution on a maximal timeinterval (0, Tmax). Let T > Tmax and assume there exists an a priori bound

R := sup(0,T )

max‖φ‖Hs+1(R2), ‖u‖Hs(R2), ‖vi‖Hs(R2).

Let tR > 0 be the maximal time of existence of solutions to the Cauchyproblem with ‖u0‖Hs , ‖φ0‖Hs+1 , ‖vi

0‖Hs ≤ R.Then there exists t ∈ (T− tR

2 , T ) such that we can consider the functionsu(x, t), vi(x, t) ∈ Hs(R2) and φ(x, t) ∈ Hs+1(R2) as initial data for a newCauchy problem with maximal time of existence T = t + tR > Tmax andwe find a contradiction .

6

DOI: 10.1685/2010CAIMXXX

We can assume also a weaker condition; as a matter of fact, to havethe continuation principle, it is enough to prove the boundness of someL∞-norms as indicated by the following lemma [17]:

Lemma 2.1. Let (W, φ) ∈ B be a solution of (3) for 0 ≤ t ≤ T , where‖u(t)‖L∞ , ‖∇φ(t)‖L∞ ≤M , then there exists a constant k(M) s.t.

‖W(t)‖Hs + ‖φ(t)‖Hs+1 ≤ c(‖W0‖Hs + ‖φ0‖Hs+1)ek(M)t, 0 ≤ t ≤ T.

2.3. Global estimates

Thanks to the continuation principle and lemma 2.1, to prove the globalexistence of solutions it is sufficient to show that the following inequalityholds true:

‖u(t)‖L∞ , ‖D1xφ(t)‖L∞ ≤M,

for every time t.This estimate is built up on sharp decay estimates obtained in [16] forthe Green function of the linearization of the hyperbolic operator and theknown decay of the heat kernel.

Given δ > 0, let us define:

(10) M δw(t) := sup

(0,t)max1, sδ‖w(s)‖L2,

(11) N δw(t) := sup

(0,t)max1, sδ‖w(s)‖L∞.

As we will prove the theorem for the lower exponent s = 2, we take(u0, v

10, v

20) ∈ H2(R2) and φ0 ∈ H3(R2). By arguing in a similar way the

existence result can be proven also for s > 2.With reference to the parabolic equation we have the following estimates:

(12)

‖φ(t)‖L∞ ≤ C(e−βt‖φ0‖L∞ +M δ0

u (t)min1, |t− 1|−δ0

),

‖D1xφ(t)‖L∞ ≤ C(e−βt‖D1

xφ0‖L∞ +M δ1

D1xu(t)min1, |t− 1|−δ1

),

‖Dkxφ(t)‖L2 ≤ C(e−βt‖Dk

xφ0‖L2 +M δk

Dkxu

(t)min1, t−δk

).

for k = 0, 1, 2.We will fix later the exponents δi. We can observe that as in the parabolicequation there is a dissipative term, these inequalities hold for δi > 0.

7

C. Di Russo et al

Thanks to the previous inequalities we obtain estimates for the functionalsdepending on φ as indicated in the following proposition:

Proposition 2.2. Let (u, v, φ) be the solution of equations (3); let M,T >

0 such that for t ∈ (0;T ) ‖u‖L∞ , ‖D1xφ‖L∞ ≤ M . Then, for t ∈ (0, T ) we

have the following estimates:

(13)

N δ0

φ (t) ≤ C(‖φ0‖L∞ +M δ0

u (t)),

N δ1

D1xφ(t) ≤ C(‖D1

xφ0‖L∞ +M δ1

D1xu(t)),

M δk

Dkxφ

(t) ≤ C(‖Dkxφ0‖L2 +M δk

Dkxu

(t)).

for k = 0, 1, 2.

From equation (8) we deduce

(14)

u (x, t) = (Γh11(t) ∗ u0)(x) + (Γh

12(t) ∗ v10)(x) + (Γh

13(t) ∗ v20)(x)

+

∫ t

0Γh

12(t− s) ∗ u(s)∂x1φ(s) + Γh13(t− s) ∗ u(s)∂x2φ(s)ds,

where the Green function Γh can be decomposed as indicated in [16] as:

Γhij = Kij + Kij +Rij .

The term Kij is the dissipative transport part which satisfies

‖DβxK(t) ∗W0‖L2 ≤ Ce−ct‖Dβ

xW0‖L2 .

The term Kij is the dissipative part for t > 1 which, in our case, is equalto:

(15) K =

[

Γp (∇Γp)t

∇Γp ∇2Γp

]

,

where Γp is the Green function of the heat equation, and Rij is a reminder.Let us start with zero order estimate for u in L2-norm:

(16)‖u(t)‖L2 ≤ ‖Γh

11(t) ∗ u0‖L2 + ‖Γh12(t) ∗ v

10‖L2 + ‖Γh

13(t) ∗ v20‖L2

+

∫ t

0‖Γh

12(t− s) ∗ u(s)∂x1φ(s)‖L2 + ‖Γh13(t− s) ∗ u(s)∂x2φ(s)‖L2ds.

8

DOI: 10.1685/2010CAIMXXX

where(17)

‖Γh11(t) ∗ u0‖L2 ≤ ‖K11(t) ∗ u0‖L2 + ‖K11(t) ∗ u0‖L2 + ‖R11(t) ∗ u0‖L2 ,

‖K11(t) ∗ u0‖L2 ≤ Ce−ct‖u0‖L2 ,

‖K11(t) ∗ u0‖L2 ≤ ‖Γp(t) ∗ u0‖L2 ≤ Cmin1, t−1

2 ‖u0‖L1 .

The remainder can be neglected, since it decays faster than the heat kernel.As for the second and third terms we have similar estimates, we just presentthe estimate of the fourth term in (16)(18)

∫ t

0‖Γh

12(t− s) ∗ u(s)∂x1φ(s)‖L2ds ≤

∫ t

0‖K12(t− s) ∗ u(s)∂x1φ(s)‖L2

+‖K12(t− s) ∗ u(s)∂x1φ(s)‖L2 + ‖R12(t− s) ∗ u(s)∂x1φ(s)‖L2ds.

By Lemma 5.2 in [16] we obtain

(19)

∫ t

0‖K12(t− s) ∗ u(s)∂x1φ(s)‖L2ds ≤

∫ t

0ce−c(t−s)‖u(s)∂x1φ(s)‖L2

≤ Cmin1, t−(δ0+δ1)M δ0

u (t)N δ1

D1xφ(t).

We estimate now the dissipative part as(20)∫ t

0‖K12(t− s) ∗ u(s)∂x1φ(s)‖L2ds =

∫ t

0‖D1

xΓp(t− s) ∗ u(s)∂x1φ(s)‖L2ds

≤

∫ t

0‖D1

xΓp(t− s)‖L2‖u(s)∂x1φ(s)‖L1ds

≤

∫ t

0Cmin1, (t− s)−1‖u(s)‖L2‖∂x1φ(s)‖L2 .

By Lemma 5.2 in [16] we obtain

(21)

∫ t

0‖K12(t− s) ∗ u(s)∂x1φ(s)‖L2ds ≤ c5t

−(δ0+δ1)M δ0

u (t)M δ1

D1xφ(t).

By arguing as in the previous part of this proof, we obtain similar esti-mates for the last term in (16).

9

C. Di Russo et al

Summing all the previous estimates we obtain:

(22)

‖u(t)‖L2 ≤ C(e−ct‖W0‖L2 + min1, t−1(‖v10‖L1 + ‖v2

0‖L1)

+ min1, t−1

2 ‖u0‖L1 + min1, t−(δ0+δ1)M δ0

u (t)N δ1

D1xφ(t)

+ t−(δ0+δ1)M δ0

u (t)M δ1

D1xφ(t)).

In a similar way we obtain the first and second order estimates for u:

(23)

‖D1xu(t)‖L2 ≤ C(e−ct‖DxW0‖L2 + min1, t−

3

2 (‖v10‖L1 + ‖v2

0‖L1)

+ min1, t−1‖u0‖L1 + min1, t−2δ1

M δ1

D1xu(t)N δ1

D1xφ(t)

+ min1, t−(δ1+δ2)N δ1

u (t)M δ2

D2xφ(t) + t−(δ0+δ1)M δ0

u (t)M δ1

D1xφ(t)),

(24)

‖D2xu(t)‖L2 ≤ C(e−ct‖D2

xW0‖L2 + min1, t−2(‖v10‖L1 + ‖v2

0‖L1)

+ min1, t−3

2 ‖u0‖L1 + min1, t−(δ2+δ1)M δ2

D2xu(t)N δ1

D1xφ(t)

+ min1, t−(δ1+δ2)N δ1

u (t)M δ2

D3xφ(t) + t−(δ0+δ1)M δ0

u (t)M δ1

D1xφ(t)).

With a similar approach we obtain the estimate of L∞-norm of u:(25)

‖u(t)‖L∞ ≤ C(e−ct‖W0‖Hs + min1, t−3

2 (‖v10‖L1 + ‖v2

0‖L1)

+ min1, t−1‖u0‖L1 + min1, t−(δ0+δ1)M δ0

u (t)M δ1

D1xφ(t)

+ min1, t−(δ1+δ2)N δ1

D1xφ(t)M δ2

D2xu(t) + min1, t−2δ1

M δ1

D1xφ(t)N δ1

u (t)

+ min1, t−(δ2+δ1)M δ2

D3xφ(t)N δ1

u (t) + min1, t−(δ1+δ0)N δ1

D1xφM

δ0

u (t)).

Thanks to the decay of the solutions of the linear problem we can nowfix the values of the different exponents in the functionals.Let us note that for the L2-norm of u in (22) the maximal decay possible

is O(t−1

2 ), while for ‖D1xu‖L2 it is O(t−1) and, for the L2-norm of the

second derivative ‖D2xu(t)‖L2 , it is O(t−

3

2 ). We can observe in (25) thatfor the L∞-norm of u the linear decay is O(t−1). Once the decay rates foru have been fixed, by proposition 2.2 we get also the decay rates for the

10

DOI: 10.1685/2010CAIMXXX

solution φ of the parabolic equation. So the functionals depending on the

L∞-norm have exponents N1

2

φ (t), N1D1

xφ(t), while the functionals depending

on L2-norm have exponents Mk+1

2

Dkxφ

(t) for k = 0, 1, 2.

Thanks to proposition 2.2 and inequalities (22)-(25) we obtain someestimates for the functionals:(26)

M1

2u (t) ≤ C(B0 +M

1

2u (t)‖D1

xφ0‖L∞ +M1

2u (t)M1

D1xu(t)

+ M1

2u (t)‖D1

xφ0‖L2 +M1

2u (t)2),

M1D1

xu(t) ≤ C(B0 +M1

2u (t)‖D1

xφ0‖L∞ +M1

2u (t)M1

D1xu(t) +M

1

2u (t)2

+ N1u(t)M1

D1xu(t) +M

1

2u (t)‖D1

xφ0‖L2 +N1u(t)‖D2

xφ0‖L2),

M3

2

D2xu

(t) ≤ C(B0 +M3

2

D2xu

(t)‖D1xφ0‖L∞ +M

3

2

D2xu

(t)M1D1

xu(t) +M1

2u (t)2

+ N1u(t)‖D3

xφ0‖L2 +N1u(t)M

3

2

D2xu

(t) +M1

2u (t)‖D1

xφ0‖L2),

N1u(t) ≤ C(B0 +M

1

2u (t)‖D1

xφ0‖L∞ +M1

2u (t)M1

D1xu(t)

+ M3

2

D2xu

(t)M1D1

xu(t) +N1u(t)‖D1

xφ0‖L2 +N1u(t)M1

D1xu(t)

+ N1u(t)‖D3

xφ0‖L2 +N1u(t)M

3

2

D2xu

(t) +M1

2u (t)‖D1

xφ0‖L2

+ M1

2u (t)M1

D1xu(t)) + ‖D1

xφ0‖L∞M3

2

D2xu

(t)).

where B0 = (‖W0‖Hs + ‖W0‖L1).Let us define

P (t) := N1u(t) +M

1

2u (t) +M1

D1xu(t) +M

3

2

D2xu

(t).

We can notice that all the previous estimates are linear combinations ofsums of type: ‖Dk

xφ0‖XFδw(t) + F δ

w(t)F δ1

w1(t) where X = L∞, L2 and

11

C. Di Russo et al

F δw, F

δ1

w1are terms of P (t).

So it is possible to estimate each of them with E0P (t) + P (t)2, whereE0 = (‖φ0‖Hs+1 + ‖D1

xφ0‖L∞).

Summing up the estimates for the different functionals yields

(27) P (t) ≤ B0 + E0P (t) + P (t)2.

If the initial data are small, this yields:

(28) P (t)2 − (1 − E0)P (t) +B0 ≥ 0.

For suitably small data, this equation implies that N1u(t), M

1

2u (t), M1

D1xu(t),

M3

2

D2xu

(t) remain bounded so ‖u(t)‖L∞ does not increase. Thanks to the

proposition 2.2 the same is true for N1

2

φ and N1D1

xφ. As we have obtained

that ‖∇φ(t)‖L∞ is bounded, from lemma 2.1 and the continuation principlewe have the global existence of smooth solutions of (3).

3. Numerical approximation and simulations.

In the previous section we proved a theorem of global existence ofsmooth solutions but unfortunately our result holds only for small initialdata. So, we are motivated to use numerical simulations as a tool to inves-tigate the evolution of solutions also for large data. One goal would be toknow whether hyperbolic system (3) has the same behavior as parabolicsystem (1), that is to say global existence for small initial data and blowup of solutions for large initial data (see [7]).It has also to be noticed that the previous analytical results about globalexistence of solutions were obtained on the whole space, whereas numericalsimulations will be performed on a bounded domain. According to whatwas proved in [11] for a one-dimensional system, we expect the convergencerates toward asymptotic state to be different: exponential decay in the caseof bounded domain with Neumann boundary conditions and polynomialdecay in the case of R

2. To match the hypotheses of Section 2, as a firststep we will carry out numerical simulations choosing initial data with com-pact support and we will observe the evolution until the solution reachesthe boundary.

Let us explain now how we solve numerically the hyperbolic-parabolicsystem (3) on a bounded domain Ω ⊂ R

2 with homogeneous Neumann

12

DOI: 10.1685/2010CAIMXXX

boundary conditions for the chemical concentration φ and the populationdensity u:

(29) ∇φ · n|∂Ω = 0, ∇u · n|∂Ω = 0,

and zero boundary condition for the normal component of v

(30) v · n|∂Ω = 0.

Let us take Ω = [0;L] × [0;L] and let us denote by δx the space step. Weconsider the discretization points xα = (α1δx, α2δx), 0 ≤ αi ≤ N + 1. Wedenote the time step by δt and the approximation of a function f at timetn = nδt by fn.For each time step, we solve first the hyperbolic equations using a relax-ation method [18] with flux limiters. The advantage of this method is theapproximation of the equations by a diagonal system, easy to be solvedand to be complemented with flux limiters. Relaxation is also a convenientsetting to extend the schemes to higher orders. Let us explain this methodin more detail.Let us denote, as in Section 2, W = (u, v1, v2) and rewrite the two firstequations of (3) as

∂tW = ∂x1A1(W) + ∂x2

A2(W) + F (W, φ),

with

A1(W) =

0 1 0γ2 0 00 0 0

W, A2(W) =

0 0 10 0 0γ2 0 0

W,

F (W, φ) =

0u∂x1

φ− v1

u∂x2φ− v2

.

We consider a simple 5-velocities relaxation scheme. Let us choose the fivevelocities as

λ1 = λ(1, 0), λ2 = λ(0, 1), λ3 = λ(−1, 0), λ4 = λ(0,−1), λ5 = (0, 0),

for some λ > 0. Now we introduce the corresponding MaxwelliansMi(W) ∈R

3, i = 1, . . . , 5, of the form

(31) Mi(W) = aiW + bi1A1(W) + bi2A2(W),

for some constants ai, bi1 and bi2.The conditions of consistency of the Maxwellians are

(32)5

∑

i=1

Mi(W) = W,

5∑

i=1

λi,jMi(W) = Aj(W), j = 1, 2.

13

C. Di Russo et al

Then, a possible choice of the coefficients ai and bij is the following

a1 = · · · = a4 = a, a5 = 1 − 4a;

b11 = b22 = −b31 = −b42 =1

2λ, bij = 0 otherwise.

It is easy to see that these coefficients satisfy conditions (32).Let us now denote by Wn,α the approximation of W at the point xα ∈ R

2

and at time tn. We set the discretization of Maxwellians (31) as

(33) fn,αi = Mi(W

n,α), for i = 1, · · · , 5.

We evolve each of the functions fi, 1 ≤ i ≤ 5, in time by following thevelocity λi :

fn+1/2,αi =fn,α

i − µ

2∑

j=1

λij(fn,αj+1i − f

n,αj−1i )

+ µ

2∑

j=1

|λij |(fn,αj+1i − 2fn,α

i + fn,αj−1i ),

where µ =δt

2δxand αj + 1 is a shift of the j-th component of the index α.

Since at least one of the component of λi is zero, each of the fi evolves inonly one direction and adding flux limiters is very simple. For example, letus write the evolution of f1 :

fn+1/2,α1 = f

n,α1 − µλ(fn,α1+1

1 − fn,α1−11 )+

µλ(

ψα1+1/2(fn,α1+11 − f

n,α1 ) − ψα1−1/2(f

n,α1 − f

n,α1−11 )

)

,

where ψα1+1/2 is an anti-viscosity term, defined as a function of the ratio

fn,α1 − f

n,α1−11

fn,α1+11 − f

n,α1

(see again, for example, [18]).

Finally, we just end by setting Wn+1 =5

∑

i=1

fn+1/2i . Here we set the veloc-

ity λ = γ and the time and space steps will have to satisfy the stability

condition γδt

δx≤ 1.

Then, we solve the parabolic equation for the chemical φ using a classicalCrank-Nicolson method for the time discretization and a Finite DifferenceMethod for the space discretization.

14

DOI: 10.1685/2010CAIMXXX

Fig. 1. Numerical simulation of the population density in (3) with two different initialconditions u0 on a square domain [0.20]× [0.20]. On the left, the initial condition, whichis a compact support perturbation of 0 and the solution at time T = 140. On the right,the initial condition, which is a compact support perturbation of the constant functionequal to 0.2 and the solution at time T = 70 are displayed.

Let us denote by M the N ×N classical second order Finite Differencematrix for the laplacian using second order derivatives for the computationof boundary values. The third equation of (3) is therefore discretized as

φn+1 − φn

δt=D

2M(φn+1 + φn) +

α

2(un+1 + un) −

β

2(φn+1 + φn),

which leads to the following linear system:(

(1 +δt

2β)I −

δt

2DM

)

φn+1 =

(

(1 −δt

2β)I +

δt

2DM

)

φn+α

2(un+1+un).

Using the above scheme, we have numerically solved system (3) withboundary condition (29), (30) on a square domain Ω = [0, 20] × [0, 20].In Figure 1 we show numerical solutions of system (3) with different ini-tial conditions. We can observe that with small initial data, like a smallperturbation of the zero state for the population density, we obtain globalexistence of solution, while if we consider large initial data the blow up ofsolution occurs.

In this regard, in [12] Hillen and Dolak proposed a model to describethe slime molds behavior, where a supplementary logistic term is intro-duced to avoid blow up. Actually the slime mold Dictyostelium discoideum

15

C. Di Russo et al

has a particular mechanism: upon starvation, the amoebae form tissue-likeaggregates. This process is controlled by chemotaxis: the cells move upwardgradients of the messenger molecule cAMP produced by the cells. Howeverthe chemotactic sensitivity may saturate and possibly vanishes for highvalues of the population density. The non-dimensional system introducedin [12] is:

(34)

∂tu+ ∇ · v = 0,

τ∂tv + γ2∇u = −v + u(1 − u)∇φ,

∂tφ = ∆φ+ αu− φ,

where u, v, and φ are used for the particle density, the particle flux, andthe signal concentration respectively. The non-dimensional system dependsonly on τ, γ and α.We can observe that it is possible to generalize the theorem of global ex-istence of smooth solutions introduced in Section 2 also for system (34).In our simulations the initial condition is a homogeneous distribution ofthe cell density with random fluctuations of 1%. Instead the flux v and thechemical concentration φ are initially zero.In Figure 2 we show the evolution of the population density for differenttimes. We notice that due to initial irregularities of the cell density, thereis a pattern formation. The aggregations continue to grow, until the satu-ration u = 1 is reached locally.

Acknowledgements.

This work has been partially supported by the project “Mathematicalproblems for the biological damage of monuments” in the CNR-CNRS 2008-2009 agreement, and PRIN project 2008-2009 “Equazioni iperboliche nonlineari e fluidodinamica”.

REFERENCES

1. C. Patlak, Random walk with persistence and external bias, Bull. Math.Biophys., vol. 15, pp. 311–338, 1953.

2. L. Keller and E. Segel, Initiation of slime mold aggregation viewed asan instability, J. Theor. Biol., vol. 26, pp. 399–415, 1970.

3. T. Hillen and K. Painter, A User’s Guide to PDE Models for Chemo-taxis, J. Math. Biol., vol. 58 (1), pp. 183–217, 2009.

4. D. Horstmann, From 1970 until present: the Keller-Segel model inchemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver., vol. 105,

16

DOI: 10.1685/2010CAIMXXX

Fig. 2. Numerical solution of system (34) with initial condition u0(x) ∈ [0.2, 0.21] on asquare domain [0.20] × [0.20] at different times T = 0, 250, 350 and 700 .

pp. 103–165, 2003.

5. A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giru-ado, G. Serini, L. Preziosi, and F. Bussolino, Percolation, morphogene-sis, and Burgers dynamics in blood vessel formation, Phys. Rev. Letters,vol. 90:118101, 2003.

6. A. Tosin, D. Ambrosi, and L. Preziosi, Mechanics and chemotaxis inthe morphogenesis of vascular networks., Bull. Math. Biol., vol. 68,pp. 1819–1836, 2006.

7. B. Perthame, Pde models for chemotactic movements: parabolic, hyper-bolic and kinetic, Appl. Math., vol. 49(6), pp. 539–564, 2004.

8. F. Filbet, P. Laurencot, and B. Perthame, Derivation of hyperbolic mod-els for chemosensitive movement, J. Math. Biol., vol. 50, pp. 189–207,2005.

9. T. Hillen and A. Stevens, Hyperbolic models for chemotaxis in 1-D,Nonlinear Anal. Real World Appl., vol. 3, pp. 409–433, 2000.

10. E. Segel, A theoretical study of receptor mechanisms in bacterial chemo-taxis, SIAM J. Appl. Math., vol. 32 no.3, pp. 653–655, 1977.

11. F. Guarguaglini, C. Mascia, R. Natalini, and M. Ribot, Stability of con-stant states of qualitative behavior of solutions to a one dimensional hy-perbolic model of chemotaxis, Discret. Contin. Dyn. S., vol. 12, pp. 39–76, 2009.

17

C. Di Russo et al

12. Y. Dolak and T. Hillen, Cattaneo models for chemosensitive move-ment. Numerical solution and pattern formation, J. Math. Biol., vol. 46,pp. 461–478, 2003.

13. C. Cattaneo, Sulla conduzione del calore, Atti del Semin. Mat. e Fis.Univ. Modena, vol. 3, pp. 83–101, 1948.

14. D. Stroock, Some stochastic processes which arise from a model ofthe motion of a bacterium, Prob. Theory and Related Fields, vol. 28,pp. 305–315, 1974.

15. T. Hillen, On the L2-moment closure of transport equations: the Cat-taneo approximation, Discr. Cont. Dyn. Systems, vol. Series B, 4(4),pp. 961–982, 2004.

16. S. Bianchini, B. Hanouzet, and R. Natalini, Asymptotic behavior ofsmooth solutions for partially dissipative hyperbolic systems with a con-vex entropy, Communications Pure Appl. Math., vol. 60, pp. 1559–1622,2007.

17. C. Di Russo, Phd thesis, work in preparation.

18. D. Aregba-Driollet and R. Natalini, Discrete Kinetic Schemes for Multi-dimensional Conservation Laws, SIAM J. Num. Anal., vol. 37, pp. 1973–2004, 2000.

18