Free Boundaries Problem for a Class of Parabolic Type Chemotaxis Model

LYU Wenbin

School of Mathematical Sciences,Shanxi University,Taiyuan 030006,China.

Abstract. In this paper,we are interested in a free boundary problem for a chemotaxis model with double free boundaries.We use contraction mapping principle and operator-theoretic approach to establish local solvability of a chemotaxis system in 1-Dimensional domain with non-constant coefficient free boundaries.

Key Words: Free boundary;chemotaxis;local solution.

1 Introduction

In this paper,we consider a free boudary problem for a chemotaxis model with double free boundaries.The model reads as follows

where

•u=u(x,t) is an unknown function of (x,t)∈(g(t),h(t))×(0,T) and it stands for the density of cellular slime molds.In other words,the densityu(x,t)occupies the domain(g(t),h(t)),an open subset of(-1,1),in timetandu(x,t)=0 in the outside of(g(t),h(t));

•v=v(x,t) is an unknown function of (x,t)∈(-1,1)×(0,T) and it stands for the concentration of chemical substances secreted by the slime molds;

•k1(x,t),k2(x,t)are given continuous functions which satisfy the Lipschitz condition onx,namely there existsL＞0,such that

for anyt∈[0,+∞).Also,k1(x,t),k2(x,t)are bounded ont∈[0,+∞).In other words,there existsC＞0 which may depend onx,such that

•g(t),h(t)are two unknown moving boundaries;

•b∈(0,1)is a given number.

For general smooth domain Ω,the system (1.1) is based on the well-known chemotaxis model with fixed boundary

introduced by E.F.Keller and L.A.Segel[1].The problem(1.4)is intensively studied by many authors(see for instance[2-8]).The initial functionsu0∈C0()andv0∈C1()are assumed to be nonnegative.Within this framework,classical results state that

•if n=1 then all solutions of(1.4)are global in time and bounded[9];

•if n=2 then

-in the caseu0(x)dx＜4π,the solution will be global and bounded[10,11],whereas

-for anym＞4πsatisfyingm∈{4kπ|k∈N}there exists initial data(u0,v0)withm=u0(x)dxsuch that the corresponding solution of (1.4) blows up either in finite or infinite time,provided Ω is simply connected[12,13];

•ifn≥3

-given anyandp＞none can find a bound onu0inLq(Ω)andv0inLp(Ω)such that(u,v)is global in time and bounded;on the other hand

-Ω is a ball then for arbitrarily small mass m ＞0 there exist u0 and v0 havingu0(x)dx=msuch that(u,v)blows up either in finite or infinite time[7].

As we all known,in a standard setting for many partial differential equations,we usually assume that the process being described occurs in a fixed domain of the space.But in the real world,the following phenomenon may happen.At the initial state,a kind of amoeba occupied some areas.When foods become rare,they begin to secrete chemical substances on their own.Since the biological time scale is much slower than the chemical one,the chemical substances are full filled with whole domains and create a chemical gradient attracting the cells.In turn,the areas of amoeba may change according to the chemical gradient from time to time.In other words,a part of whose boundary is unknown in advance and that portion of the boundary is called a free boundary.In addition to the standard boundary conditions that are needed in order to solve the PDEs,an additional condition must be imposed at the free boundary.One then seeks to determine both the free boundary and the solution of the differential equations.The theory of free boundaries has seen great progress in the last thirty years; for the state of the field we refer to[14,15].Recently,the free boundary problem has been developed rapidly in combination with economy,biology,physics and geometry.In economics,the free boundary problem can be used to study the pricing problem of Black-Schole model[16,17].In biological mathematics,free boundary value problem can be used to study the development of population and the growth of tumor[18-21].

In one dimensional case,ifkis a positive constant H.Chen and S.H.Wu[22-25]studied the similar free boundary value problem(1.1)in symmetric situation which contains only one free boundary and established the existence and uniqueness of the solution for the system (1.1).Later,high dimensional symmetry case for the free boundary value problem is considered,see [26,27].In view of the biological relevance of the particular case,nonsymmetric situation,we find it worthwhile to clarify these questions.In addition,the condition thatkis a positive constant in [23-25] seems too strict,it is also worthwhile to consider the system with non-constant coefficientk.In the present paper,we consider the system(1.1)with two free boundaries and the non-constant coefficientkin one-dimensional domain.

This paper is arranged as follows.In Section 2,we present the main result of the paper.In Section 3,we use the operator semigroup approach to establish some estimates which are essential in the proof of the main result.In Section 4,we shall give the proof of the main result.

2 Main result

Now we introduce the following space notations,which will be used in the main result here.ForT ＞0,we define

Our main result is:

Theorem 2.1.Assume p＞1and k1(x,t),k2(x,t)satisfy conditions(1.2)and(1.3).If

and

where0＜b＜1and b is a constant.Then there exist T ＞0small enough,a pair

and two curves-g(t),h(t)∈C1[0,T]∩{h|h(0)=b},which are the solutions of(1.1).

3 Some crucial estimates

In this section,we establish some crucial estimates,which will play key roles in proving the local existence of solution of the system(1.1)in one dimensional case.

3.1 Some basic properties of the solution

Lemma 3.1.If u0(x)＞0,u0∈L1(-b,b),v0∈L1(-1,1)and(u,v)is the solution of the system(1.1),then we have u(x,t)＞0,

Proof.Sinceu0(x)＞0,by standard maximal principle of the parabolic equation,it follows thatu(x,t)＞0.Integrating the first equation of(1.1)over(g(t),h(t)),we have

where the third,fourth,fifth and sixth equations of(1.1)are used.Thus one has

which implies that

as required.Integrating the eighth equation of(1.1)over(-1,1),we have

where the ninth and tenth equations of (1.1) are used.Through simple calculation,we can get

Hence,the proof of the lemma is completed.

3.2 Estimates of the solution

For any fixed-g(t),h(t)∈B,there existsT1＞T ＞0 sufficiently small such that

for anyt∈[0,T1].And let

We consider the following problems

and

Concerning the system(3.4)and(3.5),we have the following result.

Lemma 3.2.If p＞1,-g(t),h(t)∈B,

then the system(3.5)admits a unique solution

with sufficiently small T ＞0.

Proof.Step 1The simplicity of the system.

Eq.(3.5)is equivalent to the following integral equation

whereD(Δ)=H2,q(-1,1)∩{v(x,t)|vx(-1,t)=vx(1,t)=0}.

For the equation(3.4),we take the transform

and set

for-1≤ξ≤1,and(ξ,t)=0 forξ ＞1 to straighten the free boundary Γt:x=h(t).A series of detailed calculation asserts

Hence,the equation can be written as the following integral equation

The operator-theoretic feature ofeα(t,s)Δnecessary for the proof of this lemma is well known[28].There is a constantC＞0 which is independent oftsuch that

where 1≤q≤p≤+∞.

Step 2Getting the solution by contraction mapping principle.

To get the solution by the contraction mapping principle,we take

and setF(w,z)=(u,v),(w,z)∈B(M,T),whereT,M＞0 are constants and

Step 2.1F:B(M,T)(M,T).

We have

by(3.7).Using the facts(3.1),(3.2),(3.3)and(3.6),the terms of the right-hand side of(3.9)are estimated from above by

Hence,it holds that

Changing variables,we can easily get

for(w,z)∈B(L,T).If we takeM＞0 as large as

and then takeT ＞0 as small as

it holds that

We have

by(3.8).Using the facts(3.6),the terms of the right-hand side of(3.10)are estimated from above by

Hence,it holds that

for(w,z)∈B(L,T).If we takeM＞0 as large as

and then takeT ＞0 as small as

it holds that

Step 2.2Fis a contraction mapping.

For (w1,z1),(w2,z2)∈B(M,T),let (u1,v1),(u2,v2) denote the corresponding solution of the system(3.7)and(3.8)respectively.Then the difference(u1-u2,v1-v2)satisfies

i.e.

and

4 Existence of the solution

For each-g(t),h(t)∈B,we know that there exists a pair(u,v)∈X×Ysatisfying the system(3.4)and(3.5).Let

Then we know that for 0≤t1≤t2≤T,

which meansr(t)∈B.Similarly,we can show-s(t)∈B.Observe thatB ⊂C[0,T] is a compact,closed and convex subset.

DefineG:(g(t),h(t))(s(t),r(t)),thereforeGmaps-B×Binto itself.Next we will demonstrate thatGis continuous.Then Schauder theorem yields that there exist a pair(u,v)and two curves Γ:g(t),h(t)which are the solution of(1.1).

In fact,for-g1(t),-g2(t),h1(t),h2(t)∈B,let (u1,v1) and (u2,v2) represent the corresponding solutions of(3.4)and(3.5)respectively.Then for 0≤t≤T,one has

The terms on the right-hand side are estimated from above by

Now,we mainly focus on the term(I2).By Sobolev imbedding theorem and the definition of H¨older space,we have

On the other hand,we have

By parabolic theorem and Sobolev imbedding theorem,it holds that

where

Then we have

Take

It is trivial that

ash12onC[0,T].Notice

hence we have

As‖g1-g2‖C[0,T]and‖h1-h2‖C[0,T]converge to zero,(I1)and(I2)converge to zero.From this we can get that sup0≤t≤T|G(g1,h1)-G(g2,h2)|also converges to zero,which shows that the mapGis continuous onC[0,T]×C[0,T].Now the Schauder theorem yields that there exist a pair(u,v)and two curvesg(t),h(t)which are the solution of(1.1).

Obviously,gt(t)andht(t)are continuous in[0,T].

Acknowledgments

The author warmly thanks the reviewers for several inspiring comments and helpful suggestions.The research of the author was supported by the National Nature Science Foundation of China (Grant No.12101377),the Nature Science Foundation of Shanxi Province (Grant No.20210302124080) and the special fund for Science and Technology Innovation Teams of Shanxi Province(Grant No.202204051002015).