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

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 $$\overline{f}_{0}=\frac{1}{|\Omega |}\int _{\Omega }f_{0}(x)\mathrm{d}x$$ .