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        Finite-Time Blow-Up and Local Existence for Chemotaxis System with a General Memory Term

        2021-05-25 07:13:20WUYingdongWUShaohuaCHENGXinandLIULei

        WU Yingdong,WU Shaohua,CHENG Xin and LIU Lei

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

        Abstract. In this paper,we discuss the local existence of weak solutions for a parabolic system modelling chemotaxis with memory term, and we show the finite-time blowup and chemotactic collapse for this system. The main methods we used are the fixed point theorem and the semigroup theory.

        Key Words: parabolic system;chemotaxis;memory term;fixed point theorem;blow-up.

        1 Introduction

        In this paper,we consider the following model:

        where ??RN, a bounded open domain with smooth boundary??,is the unit outer normal on?? andχis a nonnegative constant.Andfis a continuous linear function and it satisfies the condition:whereLis a positive constant.

        Our model is initiated by the PKS model which is a mathematical model of biological phenomena. And this model for chemosensitive movement has been developed by Patlak,Keller and Segel[1].

        whereurepresents the population density andvdenotes the density of the external stimulus,χis the sensitive coefficient,the time constantε(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 PKS model is designed to describe the behavior of bacteria and bacteria aggregates, the question arises whether or not this model is able to show aggregation.Plenty of theoretical research uncovered exact conditions for aggregations and for blow up(see,e.g.,Childress and Percus[2,3], J¨ager and Luckhaus[4], Nagai[5], Gajewski et al.[6],Senba[7],Rasde and Ziti[8],Herrera and Velasquez[9],Othmer and Stevens[10]or Levine and Sleeman[11]).

        Global existence below these thresholds has been proven using a Lyapunov functional in Gajewski, et al. [6], Nagai, et al. [12] and Biler [13]. Besides, a number of theoretical research found exact conditions for aggregations and other properties[14–16]. Free boundary problems for the chemotaxis model are considered[17–20].

        Our study of (1.1) is also motivated by the following problem for the heat equation with a general time integral boundary condition[21]:

        where ? is a bounded domain in RNwith boundary???C1+μ(0<μ<1),is the outward normal,andu0(x)is a nonnegative function such that

        fis a nondecreasing function withf∈C1(0,∞)andf(0)>0.

        Considering the nonlinear time integral condition governing flux through the boundary,the model(1.1)involves a continuous time delay which is often referred to as a memory condition in the literature. This memory term can perfectly describe the movement of population density or the movement of single particles. Especially,the movement behavior of most species is guided by external signals: insects orient towards light sources,the smell of a sexual partner makes it favorable to choose a certain direction.

        Models with memory terms present in the boundary flux have been formulated in many applied sciences. For example, in [22], a linear memory boundary condition is introduced for the study of thermodynamics. It takes into account the hereditary effects on the boundary as those studied in [23,24]. Similar hereditary boundary conditions have been employed in models of time-dependent electromagnetic fields at dissipative boundaries[25].

        From the mathematical point of view, it is significant to study the local existence of weak solution for chemotaxis system with memory terms and the finite-time blow-up for this system.In our previous work,we have done something for this[26].

        2 Some basic lemmas

        Choose a constantσwhich satisfies

        and

        It is easy to check that (2.1) and (2.2) can be simultaneously satisfied in the case of 1≤N≤3.We define

        Here,means that0 on??}for eacht∈[0,t0]and. In the Sections 2 and 3,inessential constants will be denoted by the same letterc,even if they may vary from line to line.

        Lemma 2.1.Let p(z)be a holomorphic semigroup on a Banach space Y,with generator A.Then

        and

        Proof.The proof can be found in[27,Proposition 7.2].

        If ? is a bounded open domain with smooth boundary, on which the Neumann boundary condition is placed, then we know thatet?defines a holomorphic semigroup on the Hilbert spaceL2(?). So by Lemma 2.1,we have that

        where

        Applying interpolation to(2.3)yields

        Lemma 2.2.We assume that u∈X,a Banach space of functions,and that there is another Banach space Y such that the following four conditions hold:

        and,for some γ<1,

        Then we have a bound

        Proof.The proof can be found in[27].

        Dividing system(1.1)into two parts:

        and

        then we have the following lemmas.

        Lemma 2.3.For,t0>0small enough,problem(2.6)hasa unique solution v∈Xv,and v satisfies

        where c is a constant which is independent of T.

        Proof.It is obvious that Eq. (2.6) has a solution and the solution is unique. So what we need to proof is(2.7). LetT(t)=et?,where,then

        By(2.3),Lemma 2.2,we calculate

        Thus for small enought0,(2.7)holds.

        Lemma 2.4.For each u0∈Hσ(?)and v∈Xv,σ and N satisfy(2.1)and(2.2). The problem(2.5)has a unique solution u∈c([0,t0],Hσ(?)),and the solution can be written as

        Proof.We consider the following problem first

        Define a mapping

        whereuis the corresponding solution of(2.9).

        Then we claim that fort0small enough,G1is a contract mapping.In fact,letwe have

        By Sobolev imbedding theorems,we have

        IfN=1,

        IfN=2,3,according to(2.1)and(2.2),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(2.10),N=1,2,3,by(2.11)and(2.12),we have

        For the second term on the right side of(2.10),we have

        So we have

        which implies fort0>0 small enough,G1is contract. By Banach fixed point theorem,there exists a unique fixed point ?usuch that. Then we have the local solution of the problem(2.5):

        This completes the proof of the lemma.

        Lemma 2.5.Assume σ,N as given by(2.1)and(2.2). For solution u∈Xu of(2.5),we have

        Proof.By Lemma 2.3,(2.5)has a unique solution,and the solution can be written as

        Next we prove estimate(2.13). By(2.4)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 2.5 is proved.

        3 Local existence of solution

        In this section,we establish the local solution of system(1.1).

        Theorem 3.1.Under conditions(2.1)and(2.2),for each initial data0 on??},problem(1.1)has a unique solution(u,v)∈Xu×Xv for some t0>0.

        Proof.Considerg∈Xuandg(x,0)=u0(x) and letv=v(g) denotes the corresponding solution of the equation

        By Lemma 2.2,we havev∈Xvand

        For the solutionvof(3.1),defineu=u(v(g))to be the corresponding solution of

        Define a mapping

        Then Lemma 2.3 shows thatG2:Xu→Xu.Takeand a ball

        where the constant c is given by(2.13). Then we conclude from(2.13)and(3.2)that

        If,then fort0>0 small enoughSo fort0>0 small enough,G2mapsBMintoBM.

        Next we demonstrate that fort0small enough,G2is a contract mapping. In fact, letg1,g2∈BM?Xuandv1,v2denote the corresponding solutions of(3.1). Then

        For the first term on the right side of(3.4),

        where

        and

        Therefore

        where 0≤t≤t0.For the second term on the right side of(3.4),we have

        where

        As we have done in Lemma 2.3 and 2.4,we obtain that

        Similarly

        Then

        For the last term on the right side of(3.4),we have

        Combining the estimates(3.5),(3.6),and(3.7),it follows that

        which implies

        Consider the following equation

        By(2.7),we obtain

        Moreover,we have

        Thus fort0>0 small enough,G2is contract.

        From the process above, we have proved that problem (1.1) has a solution (u,v)∈Xu×Xvby Lemmas 2.2, 2.3 and 2.4. We derive the uniqueness by Banach fixed point theorem.

        4 Blow-up in finite time

        We then introduce an auxiliary functionF(u)defined by

        And we suppose thatN=1,then we have the following result.

        Theorem 4.1.If f(u)is a convex function on[0,∞),and F(u)satisfies

        then all nonnegative solutions of(1.1)blow up in finite time.

        Proof.In the section,without causing any confusion,we may useCi(i=0,1,2...)to denote various positive constants.

        Integrate both sides of the equation on ?,

        By using,we can get the following equality:

        Set

        By using Jensen’s inequality,we find

        Then we can get thatK(t)satisfies

        Assume to the contrary that(1.1)has a global solutionu. Then for any positive number T,we have

        Thus,by comparision,K(t)≥k(t)on[T,2T],where

        Clearly,k(t)satisfies

        Multiplying the equation in(4.2)byk′(t)and integrating fromTtot,we obtain

        Integration of this relation over(T,2T)then leads to

        where c is a positive constant chosen so thatF(c)=2F(k(T))=2F(C2). For sufficiently largeT,inequality(4.3)yields a contraction to condition(4.1),which completes the proof.

        Ackhowledgement

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

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