Existence and Asymptotic Behavior of Solutions to a Semilinear Hyperbolic-Parabolic Model of Chemotaxis

Existence and Asymptotic Behavior of Solutions to a Semilinear Hyperbolic-Parabolic Model of Chemotaxis

Abstract

We consider a general hyperbolic-parabolic model of chemotaxis in the multidimensional case. For this system we show the global existence of smooth solutions to the Cauchy problem and we determine their asymptotic behavior. Since this model does not enter in the classical framework of dissipative problems, we analyze it combining the features of the hyperbolic and the parabolic parts and using detailed decay estimates of the Green function.

Laboratoire MAPMO UMR CNRS 7349, Université d’Orléans, Fédération Denis Poisson,

UFR Sciences, Bâtiment de mathématiques, B.P. 6759 -F-45067 Orléans cedex 2, France

Keywords : Chemotaxis, hyperbolic-parabolic systems, dissipativity, asymptotic behavior.

AMS subject classifications : 35L60, 35L45, 35B40, 92B05, 92C17.

1 Introduction

Chemotaxis, the movement of cells in response to a chemical substance, is decisive in many biological processes and determines how cells arrange and organize themselves. For example, the formation of cells aggregations (amoebae, bacteria, etc) occurs during the response of the species populations to the change in the environment of the chemical concentrations. In multicellular organisms instead, chemotaxis of cell populations plays a crucial role throughout the life cycle: during embryonic development it is involved in organizing cell positioning, e.g. during gastrulation and patterning of the nervous system; in the adult life, it directs immune cell migration to sites of inflammation and fibroblasts into wounded regions to initiate healing. These same mechanisms are used during cancer growth, allowing tumor cells to invade the surrounding environment or stimulating new blood vessel growth [[15]].
The movement of bacteria under the effect of chemotaxis has been a widely studied topic in Mathematics in the last decades, and numerous models have been proposed. Moreover it is possible to describe this biological phenomenon at different scales. For example, by considering the population density as a whole, it is possible to obtain macroscopic models of partial differential equations. One of the most celebrated model of this class is the one proposed by Patlak in 1953 [[16]] and subsequently by Keller and Segel in 1970 [[13]].

In the Patlak-Keller-Segel (PKS) system, the evolution of density of bacteria is described by a parabolic equation, and the density of chemoattractant is generally driven by a parabolic or an elliptic equation. The behavior of this reaction-advection-diffusion system is now quite well-known: in the one-dimensional case, the solution is always global in time. In several space dimensions, in the parabolic elliptic case, if initial data are small enough in some norms, the solution will be global in time and rapidly decaying in time; while on the opposite, it will explode in finite time at least for some large initial data.

The simplicity, the analytical tractability, and the capacity to replicate some of the key behaviors of chemotactic populations are the main reasons of the success of this model of chemotaxis. In particular, the ability to display auto-aggregation, has led to its prominence as a mechanism for self-organization of biological systems.
Moreover, there exists a lot of variations of PKS model to describe biological processes in which chemotaxis is involved. They differ in the functional forms of the three main mechanisms involved: the sensing of the chemoattactant, which has an effect on the oriented movement of the species, the production of the chemoattractant by a mobile species or an external source, and the degradation of the chemoattractant by a mobile species or an external effect.
However, the approach of PKS model is not always sufficiently precise to describe the biological phenomena [[5]]. As a matter of fact, the diffusion can lead to fast dissipation or explosive behaviors and prevents us to observe intermediate organized structures. Moreover it is not able to reproduce the “run and tumble” behavior, the movement along straight lines, the sudden stop and the change of direction, typical of bacteria like E.Coli.

The main reason is that this approach describe processes on a long time scale, while for short time range one gets better a description from models with finite characteristic speed.
Kinetic transport equations describe quite well the movement of a single organism. For example the “run and tumble” can be described by the velocity-jump process [[8], [19]].

At an intermediate scale between diffusion and kinetic models we can find hyperbolic models. This class of models can be derived as a fluid limit of transport equations but with a different scaling, namely the hydrodynamic scaling , [[2]].
Starting from a transport equation for the chemosensitive movements, in [[7]] Hillen shows a kinetic derivation of hyperbolic models by the moment closure method, thus obtaining the Cattaneo model for chemosensitive movement. Using the first two moments he obtains the following hyperbolic-parabolic model:

(1)

where , , is the population density, are the fluxes, is the concentration of chemical species, and the source terms are smooth functions.

We start our analytical study by considering the semilinear hyperbolic-parabolic system

(2)

which generalize the one proposed by Dolak and Hillen in [[4]]. The parameter is the characteristic speed of propagation of the cells and the source terms , and are smooth functions.

The coupling of hyperbolic and parabolic equations has been widely studied by Kawashima and Shizuta [[11], [12], [17]]. Under the smallness assumption on the initial data and the dissipation condition on the linearized system, they were able to prove global (in time) existence and asymptotic stability of smooth solutions to the initial value problem for a general class of symmetric hyperbolic-parabolic systems.
System (1) does not enter in this framework. As a matter of fact, due to the presence of the source term , the dissipative condition fails.
With reference to the one dimensional case, a first result of local and global existence for weak solutions, under the assumption of turning rate’s boundness, was proved in [[10]]. Subsequently Guarguaglini et al. in [[6]] have proved more general results for this model under weaker hypotheses, by showing a general result of global stability of zero constant states for the Cauchy problem and of general constant state for the Neumann problem. These results have been obtained using the linearized operators, and the accurate analysis of their nonlinear perturbations.

In order to obtain our global existence result for the multidimensional case we follow this approach. The basic idea is to consider the hyperbolic and parabolic equation “separately”, and to take advantage of their respective properties.

Thanks to the Green function of the heat equation , and the Duhamel’s formula, we know that, the solution to the parabolic equation is:

On the other hand for the damped wave equation,

(3)

we have, by the theory of dissipative systems [[18]], that the presence of the dissipative term enforces a faster decay of the solution. This implies that we can write the solution of the hyperbolic part of system (1), as

where , and is the Green function of the damped wave equation (3).

Our strategy has been to use the decomposition of the Green function of dissipative hyperbolic systems done by Bianchini at al. [[1]] and its precise decay rates.
Indeed in [[1]] the authors proposed a detailed description of the multidimensional Green function for a class of partially dissipative systems. They analyzed the behavior of the Green function for the linearized problem, decomposing it into two main terms. The first term is the diffusive one, and consists of heat kernel, while the faster term consists of the hyperbolic part. Moreover they gave a more precise description of the behavior of the diffusive part, which is decomposed into four blocks, which decay with different decay rates. They showed that solutions have canonical projections on two different components: the conservative part and the dissipative part. The first one, which formally corresponds to the conservative part of the equations, decays in time like the heat kernel, since it corresponds to the diffusive part of the Green function. On the other side, the dissipative part is strongly influenced by the dissipation and decays at a rate faster than the conservative one.
By these refined estimates we were able to prove global existence of smooth solutions for small initial data, and to determine at the same time their asymptotic behavior.

We are able to show decay rates of the -norm of solution of order , faster than the one obtained in [[6]] which was .
Moreover we show the global existence, and we determinate the asymptotic behavior of solutions, also for perturbation of non-zero constant stationary states in the case of simpler source terms. In order to prove this result, we need to adapt the decay estimates of the Green function to compensate the lack of polynomial decay of linear term in the hyperbolic equations.

The parabolic and hyperbolic models of chemotaxis are expected to have the same behavior for long time. We investigate this aspect analytically and we show that the difference between the solution of PKS model and the hyperbolic one decays with a rate of in , so faster than the decay of solutions themselves if , otherwise we get a decay faster than the decay of solutions.

The article is organized as follows: in the first section, we review some properties of partially dissipative hyperbolic systems, we recall the results obtained by Bianchini et al. in [[1]] about the asymptotic behavior of their smooth solutions, and the local existence in time for smooth solutions to system (2) to the Cauchy problem. Subsequently, in Section 3, we are able to prove the global existence result thanks to the refined decay estimates of the Green Kernel of hyperbolic equations.
In Section 4 we study the case of perturbation of non-zero constant stationary state. For large time hyperbolic and parabolic model are expected to have the same behavior. Then, in the last section,

we examine the difference between solutions to the hyperbolic-parabolic system (2) and to the related PKS model, showing that this difference decays with a faster rate.

2 Background

2.1 Partially Dissipative Hyperbolic Systems

In this first section we recall some properties of hyperbolic dissipative systems.
Let us focus our attention on the following multidimensional system of balance laws

(4)

where , , with initial conditions

We can observe that since (4) is equivalent to damped wave equation, the behavior of the solutions to the Cauchy problem for this system is quite well known [[11]]. Moreover system (4) belongs to the class of dissipative hyperbolic systems.

It is possible to rewrite system (4) in a compact form as

(5)

where , and

with , , , and and is the canonical th vector of . Here we denote the source term by

The initial condition reads

(6)

By the introduction of new variables , with

and a symmetric positive definite matrix , defined as

(7)

it is possible to symmetrize system (5). Selecting as new variable, our system reads

where

and . Let us notice that, for every , the matrix is symmetric.

In order to continue the analysis of smooth solutions for dissipative hyperbolic system let us introduce the condition of Shizuta and Kawashima (SK) [[17]] for hyperbolic systems.

Definition 2.1.

System (5) verifies condition (SK), if every eigenvector of is not in the null space of for every .

We can observe that system (5) verifies the Kawashima condition since, given an equilibrium state

then the generic vector

is eigenvector of , if and only if .

With reference to the existence of smooth solutions to system (5), we recall the following result, which is a special case of the results in [[1]].

Theorem 2.2.

Let us consider the Cauchy problem (5)-(6). Let . For every , there is a unique global solution to (5)-(6) which verifies

and such that,

where is a positive constant.

The refined estimates of the Green Kernel of system (5) proposed by Bianchini et al. [[1]], holds for linearized dissipative system in the Conservative-Dissipative form. Then, we rewrite system (5) in this particular form, which will be useful in our study.
Let us consider a linear system with constant coefficients

(8)

where .

Definition 2.3.

System (8) is in Conservative-Dissipative form (C-D form) if it is symmetric, i.e. for all , and there exists a negative definite matrix , such that

In this case is called the conservative variable, while is the dissipative one.
Under suitable assumptions every symmetrizable dissipative system can be rewritten in the C-D form. Let us observe that system (5) can be easily written in the Conservative-Dissipative form by a change of variable.

Set

and define the matrices of the C-D form

Setting

and reporting in (4), we obtain the conservative-dissipative form for this system

We will consider by now the Conservative-Dissipative form of system (4) written as:

(9)

The Multidimensional Green Function

We present now the results on the study of the Green Kernel of multidimensional dissipative hyperbolic systems done by Bianchini et al. in [[1]]. In their work the authors analyzed the behavior of the Green function for linearized problems, which has been decomposed into two main terms. The first term, the diffusive one, consists of heat kernels, while the faster term consists of the hyperbolic part.
In general, the form of the Green function is not explicit, but it is possible to deal with its Fourier transform. The separation of the Green kernel into various parts is done at the level of a solution operator acting on .

They proved the following theorem, [[1]]:

Theorem 2.4.

Consider the linear PDE in the conservative-dissipative form

where , satisfy the assumption (SK), and let , be the eigenprojectors on the null space and the negative definite part of with and .

Then, for any function the solution of the linear dissipative system can be decomposed as

where for any multi index and for every , the following estimates hold.
estimates:

estimates:

2.2 Local Existence of Smooth Solutions

Since our aim is to prove the global existence of smooth solutions with small initial data to the complete hyperbolic-parabolic system (2), a sharp results of local existence of solutions in essential for our proof. Let us consider a more general semilinear hyperbolic-parabolic system

(10)

where , , and , with . We complement the system with the initial conditions

(11)

and with the regularity assumptions

(12)

With reference to the local existence of smooth solutions to system (10), we recall the following result:

Theorem 2.5.

There exists , only depending on initial data, such that, under the assumptions that, for , are locally Lipschitz maps, problem (10)-(11)-(12), has a unique local solution

This theorem can be proved with a standard fixed point method [[3]], and it is a special case of Theorem 2.9 in [[11]].

3 The Cauchy Problem

At the beginning of this section we recall some results which will be useful to establish the existence of global solutions to the more specific problem

(13)

with the initial conditions

(14)

and the regularity assumptions

(15)

In order to prove our results we make some assumptions on the functions on the right hand side in system (13).

():

and

where , and for all fixed

where is a suitable constant depending on .

():

and . In particular for all fixed with

where is a suitable constant depending on .

():

and . For all fixed with

where is a suitable constant depending on .

Let us notice that under the assumptions this general sensitivity function, , covers different possible relations between species and chemical substance present in chemotaxis models as reported in [[9]].

():

and

where , and for all fixed ,

where is a suitable constant depending on .

By these assumptions, we are led to consider the system

(16)

It is possible to rewrite the above system as

(17)

where

and

Thanks to the regularity of source terms, the local Lipschitz condition yields. Then we can apply Theorem 2.5 and deduce the local existence of solution to (16).
Before proceeding in our study of global existence of solutions we recall some well-known inequalities in the Sobolev spaces [[20]].

Proposition 3.1.

Let , then

If ,

if , then

Proposition 3.2.

Let be smooth and assume . Then, for

Proposition 3.3.

Let such that there exists that for ,

Then for every smooth function

with , .

3.1 Continuation Principle

Now we are going to prove the existence of global solutions to problem (16)-(14)-(15) using the following Continuation Principle.

Proposition 3.4.

Let be the maximal time of existence for a local solution to system (16)-(14)-(15). Then

Proof.

Let be a given local smooth solution on a maximal time interval .
Let and assume there exists an a priori bound

Let be the maximal time of existence of solutions to the Cauchy problem, with , . Then, there exists such that, we can consider the functions and as initial data for a new Cauchy problem, with maximal time of existence , and we find a contradiction. ∎

From the previous result, it is enough to estabilish an a priori , bound to give the global existence. Beside we can notice that to prove the global existence result, it is enough to prove the boundness of -norm of functions , as showed by the following Lemma.

Lemma 3.5.

Let a solution of (16) for , where , , then there will exist a constant such that,

Proof.

Let , then we want to prove that norms of these functions are bounded.
Thanks to the Duhamel’s formula we can write the solution of the hyperbolic part as

where is the Green function of system (9). Then

by Proposition 3.1 we deduce

Let us observe that, by assumptions ,,, and Proposition 3.3, we have

By previous inequalities we get

The last relation can be written as:

(18)

where the constant depends on and , .
Let us consider now the solution of the parabolic equation, that thanks to Duhamel’s formula we can write as

Then, we can estimate the -norm of as follows

where the constant depends on and . If we sum the last inequality and (18) we obtain

Applying Gronwall’s Lemma we easily deduce

(19)

3.2 Global Existence and Asymptotic Behavior of Smooth Solutions

In this section our aim is to prove the boundness of solutions to system (16) for every time . Once that this result will be obtained, we could easily prove the global existence of solutions by Lemma 3.5 and Continuation Principle 3.4. The estimates are built up on sharp decay estimates, obtained by Theorem 2.4 for the Green function of the hyperbolic operator and the known decay of the heat kernel.
Let us observe that by this approach, we get simultaneously the boundness of norm of solutions and also their decay rates. Given , let us define for a given function the functionals

Moreover let us denote by any space derivative , such that .
Before starting our proof, let us recall an useful lemma [[1]]:

Lemma 3.6.

For any ,

it holds

Decay Estimates for the Chemoattractant

We can collect the estimate referred to the function in the following proposition.

Proposition 3.7.

Let be the solution of system (16)-(14)-(15), under the assumptions , , , . Let such that for , , . Then for ,