Asymptotic behavior of a quasilinear parabolic–elliptic–elliptic chemotaxis system with logistic source

Abstract

In this paper, we study the following quasilinear parabolic–elliptic–elliptic chemotaxis system with indirect signal production and logistic source

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot (D(u)\nabla u) -\nabla \cdot (S(u)\nabla v)+\mu (u-u^\gamma ) ,&\qquad \quad x\in \Omega ,\,t>0,\\&0=\Delta v- v+ w,&\qquad \quad x\in \Omega ,\,t>0,\\&0=\Delta w- w+ u,&\qquad \quad x\in \Omega ,\,t>0 \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset \mathbb {R}^n(n\ge 1)$, where $\mu>0, \gamma >1$, and $D, S\in C^2\,([0,\infty ))$ fulfilling $D(s)\ge a_0(s+1)^{\alpha },\, |S(s)|\le b_0s(s+1)^{\beta -1}$ for all $s\ge 0$ with $a_0, b_0>0$ and $\alpha ,\beta \in \mathbb {R}$ are constants. The purpose of this paper is to prove that if $\beta \le \gamma -1$, the nonnegative classical solution (u, v, w) is global in time and bounded. In addition, if $\mu >0$ is sufficiently large, the globally bounded solution (u, v, w) satisfies

$$\begin{aligned} \Vert u(\cdot ,t)-1\Vert _{L^\infty (\Omega )}+\Vert v(\cdot ,t)-1\Vert _{L^\infty (\Omega )}+\Vert w(\cdot ,t)-1\Vert _{L^\infty (\Omega )} \rightarrow 0 \quad as \quad t\rightarrow \infty . \end{aligned}$$