Existence of Solutions for a Parabolic System Modelling Chemotaxis with Memory Term

WU Shaohua and LIU Lei

School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China.

Abstract.In this paper,we come up with a parabolic system modelling chemotaxis with memory term,and establish the local existence and uniqueness of weak solutions.The main methods we use are the fixed point theorem and semigroup theory.

Key Words:Parabolic system;chemotaxis;memory term;fixed point theorem.

1 Introduction

We consider in this paper a parabolic-parabolic system modelling chemotaxis,which explains the movement of population density or the movement of single particles. The movement behavior of many species is guided by external signals,for instance,amoebas move upwards chemical gradients,insects orient towards light sources,the smell of a sexual partner makes it favorable to choose a certain direction.

The basic model of chemotaxis was introduced by Keller and Segel in[1],and this reads

whereurepresents the population density andvdenotes the density of the external stimulus,χis the sensitive coefficient,the time constant 0≤ε≤1 indicates that the spatial spread of the organismsuand the signalvare on different time scales.The caseε=0 corresponds to a quasi-steady state assumption for the signal distribution.

Since the KS model is designed to describe the behavior of bacteria,the question arises whether or not this model is able to show aggregation.A number of theoretical research found exact conditions for aggregations and for blow-up[2–16].Besides,free boundary problems for the chemotaxis model is considered[17–24].

The possibility of blow-up has been shown to depend strongly on space dimension.For instance,in the case ofχis a constant,andg(u,v)=-γv+αu,finite time blow-up never occurs in 1-D case[25](unless there is no diffusion of the attractantv,[26,27]),but can always occur in N-D cases forN≥3.For the 2-D case,it depends on the initial data,i.e.there exists a threshold;if the initial distribution exceeds its threshold,then the solution blows up in finite time,otherwise the solution exists globally[28].

We introduce a memory term in equation(1.1),i.e.

wherep,q≥0 are nonnegative constants.

There are two main sources of motivation for our investigation of(1.3).One of them is the semilinear heat equation model with memory,

and the other is

It is known that the solutions of(1.4)and(1.5)are global in the case of 0≤p+q≤1[29,30].

Letg(u,v)=-γv+αu,χ(v)=χis a constant,and supplement equations(1.3),(1.2)with some initial boundary conditions,then we obtain the following system

Standard notationHm(Ω)denotes Sobolev spaceWm,2(Ω)throughout this paper.Inessential constants will be denoted by the same letter c,even if they may vary from line to line.

2 Main results

Considering the case ofp=1,q=0,ε=1,and for simplicity sake,we might as well assumeγ=α=1,then problem(1.6)turns into

where Ω ⊂RN,a bounded open domain with smooth boundary∂Ω,nis the unit outer normal on∂Ω andχis a nonnegative constant.

Choose a constantσwhich satisfies

and

It is easy to check that(2.2)and(2.3)can be simultaneously satisfied in the case of 1≤N≤3.Our main results are as follows.

Theorem 2.1.Under conditions(2.2)and(2.3),for each initial dataproblem(2.1)has a unique local solution(u,v)∈Xu×Xv for somet0＞0.

We define here

3 Some basic lemmas

Lemma 3.1.Let p(z)be a holomorphic semigroup on a Banach space Y,with generator A.Then

and

Proof.Refers to[31]Proposition 7.2.

If Ω is a bounded open domain with smooth boundary,on which the Neumann boundary condition is placed,then we know thatetΔdefines a holomorphic semigroup on the Hilbert spaceL2(Ω).So by Lemma 3.1,we have that

where

Applying interpolation to(3.1)yields

Divide system(2.1)into two parts:

and

Then we have the following lemmas.

Lemma 3.2.Fort0＞0small enough,problem(3.4)hasa unique solution v∈Xv,and v satisfies

where c is a constant which is independent of T.

Proof.It is obvious that equation(3.4)has a solution and the solution is unique.So what we need to proof is(3.5).Letwherethen

By(3.2)we calculate

Thus for small enought0,(3.5)holds.

Lemma 3.3.If u0∈X and v∈Xv,then problem(3.3)has a unique solution u∈Xu,and

Proof.We consider the following problem first

Define a mapping

whereuis the corresponding solution of(3.7).

Then we claim that fort0small enough,G1is a contract mapping.In fact,letandu1,u2denote the corresponding solutions of(3.7),we have

By Sobolev imbedding theorems,we have

IfN=1,

IfN=2,3,according to(2.2)and(2.3),we obtain that

which implies

Hence forN=1,2,3,we have

Similarly,we have

So for the first term on the right side of(3.8),N=1,2,3,by(3.9)and(3.10),we have

For the second term on the right side of(3.8),we have

So we have

which implies fort0＞0 small enough,G1is contract.By Banach fixed point theorem,there exists a unique fixed pointsuch thatso(3.3)has a unique solution,and the solution can be written as

Next we prove estimate(3.6).By(3.2)we have

By Sobolev imbedding theorem,forN=1,we have

ForN=2,3,we have

So we obtain that,for 0≤t≤t0,

Meanwhile,we deduce

and

Hence we declare that

which implies fort0small enough

Thus,Lemma 3.3 is proved.

4 Local existence of solution

In this section,we establish the local solution of system(2.1).

Theorem 4.1.Under conditions(2.2)and(2.3),for each initial data0on ∂Ω},problem(2.1)has a unique solution(u,v)∈Xu×Xv for some t0＞0.

Proof.Considerw∈Xuandw(x,0)=u0(x)and letv=v(w)denotes the corresponding solution of the equation

By Lemma 3.2,we havev∈Xvand

For the solutionvof(4.1),defineu=u(v(w))to be the corresponding solution of

Define a mapping

then Lemma 3.3 shows thatG2:Xu→Xu.

Takeand a ball

where the constant c is given by(3.6).Then we conclude from(3.6)and(4.2)that

Ifthen fort0＞0 small enough.So fort0＞0 small enough,G2mapsBMintoBM.

Next we demonstrate that fort0small enough,G2is a contract mapping.In fact,letw1,w2∈BM⊂Xuandv1,v2denote the corresponding solutions of(4.1).Then

For the first term on the right side of(4.4),

where

and

Therefore,

For the second term on the right side of(4.4),we have

where

As we have done in Lemma 3.3,we obtain that

Similarly,

Then

For the last term on the right side of(4.4),we have

Combining the estimates(4.5),(4.6),and(4.7),it follows that

which implies

Consider the following equation

By(3.5),we obtain

What’s more,

Thus fort0＞0 small enough,G2is contract.

From the process above,we have proved that problem(2.1)has a solution(u,v)∈Xu×Xvby Lemma 3.2 and 3.3.We derive the uniqueness by Banach fixed point theorem.

Acknowledgement

We would like to thank the referees for carefully reading the manuscript and for their helpful suggestions.