Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source

Abstract

This paper is devoted to studying the following quasilinear parabolic-elliptic-elliptic chemotaxis system \begin{equation*} \begin{cases} u_{t}=\nabla\cdot(\varphi(u)\nabla u-\psi(u)\nabla v)+au-bu^{\gamma},\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0=\Delta v-v+w^{\gamma_{1}}, \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0=\Delta w-w+u^{\gamma_{2}}, \ &\ \ x\in \Omega, \ t>0 , \end{cases} \end{equation*} with homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega\subset\mathbb{R}^{n}(n\geq 1),$ where $a,b,\gamma_{2}>0, \gamma_{1}\geq 1, \gamma>1 $ and the functions $\varphi,\psi\in C^{2}([0,\infty)$ satisfy $\varphi(s)\geq a_{0}(s+1)^{\alpha}$ and $|\psi(s)|\leq b_{0}s(1+s)^{\beta-1}$ for all $s\geq 0$ with $a_{0},b_{0}>0$ and $\alpha,\beta \in \mathbb{R}.$ It is proved that if $\gamma-\beta\geq \gamma_{1}\gamma_{2} ,$ the classical solution of system would be globally bounded. Furthermore, a specific model for $\gamma_{1}=1,\gamma_{2}=\kappa$ and $\gamma=\kappa+1$ with $\kappa>0$ is considered. If $\beta\leq 1$ and $b>0$ is large enough, there exist $C_{\kappa},\mu_{1},\mu_{2}>0$ such that the solution$(u,v,w)$ satisfies \begin{align*} \left\|u(\cdot,t)-\left(\frac{b}{a}\right)^{\frac{1}{\kappa}}\right\|_{L^{\infty}(\Omega)}+\left\|v(\cdot,t)-\frac{b}{a}\right\|_{L^{\infty}(\Omega)}+\left\|w(\cdot,t)-\frac{b}{a}\right\|_{L^{\infty}(\Omega)} \leq \begin{cases} C_{\kappa}\mbox{e}^{-\mu_{1}t}, \ &\ \ \mbox{if} \ \kappa \in (0,1], \\[2.5mm] C_{\kappa}\mbox{e}^{-\mu_{2}t}, \ &\ \ \mbox{if} \ \kappa \in (1,\infty), \end{cases} \end{align*} for all $t\geq 0.$ The above results generalize some existing results.

Keywords

• chemotaxis system
• nonlinear indirect secretion
• global boundedness
• long time behavior