Asymptotic Behavior in a Quasilinear Fully Parabolic Chemotaxis System with Indirect Signal Production and Logistic Source

LI Dan and LI Zhongping

College of Mathematic and Information,China West Normal University,Nanchong 637009,China.

Abstract. In this paper,we study the asymptotic behavior of solutions to a quasilinear fully parabolic chemotaxis system with indirect signal production and logistic sourceunder homogeneous Neumann boundary conditions in a smooth bounded domainΩ ⊂Rn (n ≥1),where b ≥0, γ ≥1, ai ≥1, µ,bi>0 (i=1,2), D,S ∈C2([0,∞))fulfilling D(s)≥a0(s+1)−α,0 ≤S(s)≤b0(s+1)β for all s ≥0,where a0,b0>0 and α,β ∈R are constants.The purpose of this paper is to prove that if b ≥0 and µ>0 sufficiently large,the globally bounded solution(u,v,w)with nonnegative initial data(u0,v0,w0)satisfies

Key Words:Chemotaxis system;indirect signal;logistic source;asymptotic behavior.

1 Introduction

In 1970,Keller and Segel proposed a classical biological chemotaxis model[1]

withurepresents the density of cells,vis the density of a chemical signal.In the first equation of(1.1),∆udenotes self diffusion and the cross-diffusion term −∇·(u∇v)means that the cell is moving towards a high chemical concentration.In the second equation of(1.1),∆vis the self-diffusion of the chemical signal,−v+udenotes the consumption ofvand direct production by the cellu.The system(1.1)describes the chemotactic behavior of cells in numerous biological processes[2,3],and the biological model(1.1)plays a key role.Whenn=1,the system(1.1)has a unique global solution[4].Whenn=2,there is a critical mass phenomenon[5],if,the system(1.1)processes a globally bounded classical solution;if,the solution of the system(1.1) will blow up [6].Whenn≥3,if Ω is a ball,then for arbitrarily small massm:=there exists(u0,v0)such that(u,v)blowing up[7].

Ever since 1970,mathematicians have intensively investigated different types of chemotaxis models for a variety of chemotaxis processes [2].When considering the logistic source,some researchers studied the corresponding quasilinear chemotaxis system of(1.1)

withD,S∈C2([0,∞))satisfyingD(s)≥c0sρ,c1sq≤S(s)≤c2sq.Whenf≡0,previous works have investigated whether the solutions are globally bounded or blow up [8–10].It is well-known that logistic sources favor the existence of the globally bounded solution.Indeed,iff(s) is a smooth function fulfillingf(0)≥0 andf(s)≤as−µs2for alls>0,it is shown that wheneverq<1,there exists a unique globally bounded and classical solution [11,12].Whenf(s)≤a−µs2for alls≥0,witha≥0 andµ>0 properly large,ifn≥3 and Ω is convex,Winkler [13] showed the global boundedness of solutions.The chemotaxis signal of(1.2)is produced directly by cells,yet the signal generation undergoes intermediate stages in some realistic biological processes[14–16],the indirect signal production mechanism can cause different interaction of cross-diffusion and the logistic source.The chemotaxis system with indirect signal production[16]was introduced

withµ>0,p>1.It was shown that ifthe solution is globally bounded;ifµ>0 is sufficiently large,(u,v,w)satisfies‖u(·,t)−1‖L∞(Ω)+‖v(·,t)−1‖L∞(Ω)+‖w(·,t)−1‖L∞(Ω)→0 ast→∞.

Wang[17] studied the more realistic nonlinear diffusion situation and came up with the following quasilinear fully parabolic chemotactic system with indirect signal production and logistic source

in a smooth bounded domain Ω ⊂Rn(n≥1) with homogeneous Neumann boundary conditions,where the parametersb≥0,γ,ai≥1,µ,bi(i=1,2)are assumed to be positive constants,νdenotes the outer normal derivative on∂Ω,the functionsD,S∈C2([0,∞))satisfy

wherea0,b0>0 andα,β∈R are constants.The author considered the boundedness of solutions in (1.4),then he obtained the global boundedness of solutions in four cases:the self-diffusion dominates the cross-diffusion;the logistic source suppresses the crossdiffusion;forµ>0 sufficiently large,the logistic dampening balances the cross-diffusion;the self-diffusion and the logistic source both balance the cross-diffusion withµ>0 suitably large.

Inspired by the above works[15–17],in this paper,we further consider the asymptotic behavior of solutions to(1.4).

The main result of this paper reads as follows.

Theorem 1.1.Let γ≥1,Ω⊂Rn(n≥1)be a bounded domain with smooth boundary.Assume that D,S satisfy(1.5)and(1.6)with a0,b0>0,α,β∈R.Suppose that any nonnegative initialvalue0<ω<1.Then there exist b≥0andµ>0properly large,such that the global classical solution of(1.4)satisfies

This paper is organized as follows.In Section 2,we obtain a lower bound foruby setting up an estimate of ∆vand using the comparison principles.In Section 3,we construct the corresponding energy function to prove Theorem 1.1.

2 A lower bound for u

To prove the main result,at the beginning,we give a lemma about the Laplacian in Ωwith homogeneous Neumann boundary conditions.

Lemma 2.1(Neumann Laplacian[16,18]).Let p∈(1,∞),λ∈(0,1).∆N shows the realization of the Laplacian in Lp(Ω)with domain

Then the operator A:=−∆+λ is sectorial and processes closed fractional powers Aδ for arbitrary δ>0with dense domain D(Aδ)known to have the embedding property

Furthermore,if(e−tA)t≥0represents the corresponding analytic semigroup,we can see that there exists a constant Cp,δ>0such that

provided ϕ∈Lp(Ω).

Inspired by[19],then we establish an estimate of ∆vpromising the validity of a lower bound foru.

Lemma 2.2.Let γ≥1,δ0∈(0,1)and a2≥1.If(u0,v0,w0)∈Cω×W1,∞(Ω)×W1,∞(Ω),0<ω<1,and(u,v,w)is the solution of(1.4).Then for some p>1,there exists a constant C1>0independent of initial data such that

Proof.According to the globally bounded solution of(1.4),there existsC2such that

for allt>0.Then there exists suitably larget0=t0(u,v,w)>0 satisfying

For the third equation in(1.4),by means of the variation-of-constants representation,we have

for allt∈(t0,∞).In view of(2.2)and(2.4),we can see that

for allt∈(t0,∞),whereC∗P,δ0>0 anddσ<∞on the condition thatδ0∈(0,1)anda2≥1.Thereafter,

with someC1>0.

Lemma 2.3.Let γ≥1,a1≥1.If(u0,v0,w0)∈Cω×W1,∞(Ω)×W1,∞(Ω),0<ω<1,and(u,v,w)is the solution of(1.4),we can choose a constant C∗>0independent of initial data such that

Proof.According to(2.3),then there existst0=t0(u,v,w)>0 satisfying

In addition,let us fix anyδ1∈(1,2) satisfyingδ1−1<δ0<1 and choosep>1 satisfyingthen

Employing the variation-of-constants representation forv,we obtain

for allt≥t0.Combining(2.1),(2.6)and(2.7),we can write

for allt∈(t0,∞),wheredσ<∞on the condition thatδ1−δ0∈(0,1)anda1≥1.Thereafter,

with someC∗>0.

Lemma 2.4.Let γ≥1.If(u0,v0,w0)∈Cω×W1,∞(Ω)×W1,∞(Ω),0<ω<1,and(u,v,w)is the solution of(1.4).Assume that D,S satisfy(1.5)and(1.6),then there exists a constant b∗>0such that if b>b∗,thus

Proof.According to(2.5),we can see that there existst0=t0(u,v,w)>0 satisfying

Letb>b∗:=max{2γC∗b0,2βC∗b0+2β−1µ},we shall treat the first equation of(1.4)in two cases:γ≥βandγ<β.

Whenγ≥β,the first equation of(1.4)infers that

for allx∈Ω,t≥t0.Assume that the functiony(t)∈C1([t0,∞))satisfies

Therefore,relying on the comparison principle[19],we can write

Whenγ<β,the first equation in(1.4)infers that

for allx∈Ω,t≥t0.Suppose that the functiony(t)∈C1([t0,∞))satisfies

Therefore,using the comparison principle,we can see that

3 Asymptotic behavior

First,the following lemma from elementary analysis[20]is a useful tool in this section.

Lemma 3.1.Let f:(1,∞)→[0,∞)be uniformly continuous such thatThen

We next construct an appropriate energy function to(1.4),the energy function is prepared for the proof of asymptotic behavior.According to the parabolic regularity [21]with the global boundedness of(u,v,w),we conclude that there existρ∈(0,1)andH>0 such that

for allt>1.We letψ:(0,∞)→R be defined by

Lemma 3.2.Let γ≥1.If(u0,v0,w0)∈Cω×W1,∞(Ω)×W1,∞(Ω),0<ω<1,and(u,v,w)is the solution of(1.4),and D,S satisfy(1.5)and(1.6).Then there exist b≥0andµ>0properly large,such that the functions E(t)and F(t)satisfy

Proof.Notice thatE(t)≥0 is obvious.In terms of the first equation in(1.4),we obtain

In terms of the globally bounded solution of(1.4),we can see that(u+1)2β+α≤Kfor eachα,β∈R,t>0.Using integration by parts,the mean value theorem,Young’s inequality and(u+1)2β+α≤K,we have

withθ=θ(x,t)∈(0,1) for allx∈Ω andt>0.Using the second equation of(1.4),we can see that

for allt>0.Using the third equation in(1.4),we can write

for allt>0.Hence,collecting terms above(3.5)-(3.7),we get

In terms of(3.8),using Young’s inequality and(2.8),we have

withθ=θ(x,t)∈(0,1)for allx∈Ω andt>0,we could split into two casesγ≥2 and 1≤γ<2 to arrive at the estimate.

Ifγ≥2,in light of Lemma 2.4,there existst0>0 suitably large such that

withx∈Ω,t≥t0for allµ>µ1.

If 1≤γ<2,in light of(2.4)and Lemma A.1[22],there existsµ1>0 suitably large such thatM(x,t)≥1/4.Combining with(3.9),we complete the proof of(3.4).

We are now in the position to prove our main result.

Integrating(3.10)fromt0to ∞,we obtain

Acknowledgments

The paper is supported by the National Science Foundation of China(11301419)and the Meritocracy Research Funds of China West Normal University[17YC382].