Effects of density-suppressed motility in a two-dimensional chemotaxis model arising from tumor invasion

Abstract

This paper deals with the global stability of the following density-suppressed motility system

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\Delta (\varphi (v)u), &{} x\in \Omega ,\quad t>0,\\ v_{t} =\Delta v+wz, &{} x\in \Omega ,\quad t>0,\\ w_{t}=-wz, &{} x\in \Omega ,\quad t>0,\\ z_{t}=\Delta z-z+u, &{} x\in \Omega ,\quad t>0 \end{array} \right. \end{aligned}$$

in a bounded domain $\Omega \subset \mathbb {R}^{2}$ with smooth boundary, where the motility function $\varphi (v)$ is positive. If $\varphi (v)$ has the lower-upper bound, we can obtain that this system possesses a unique bounded classical solution. Moreover, we can obtain that the global solution (u, v, w, z) will exponentially converge to the equilibrium $(\overline{u}_{0},\overline{v}_{0}+\overline{w}_{0},0,\overline{u}_{0})$ as $t\rightarrow +\infty$, where $\overline{f}_{0}=\frac{1}{|\Omega |}\int _{\Omega }f_{0}(x)\mathrm{d}x$.