Asymptotic behavior of a quasilinear Keller--Segel system with signal-suppressed motility

Abstract

This paper is concerned with the density-suppressed motility model: $u_{t}=\Delta (\displaystyle\frac{u^m}{v^\alpha}) +\beta uf(w), v_{t}=D\Delta v-v+u, w_{t}=\Delta w-uf(w)$ in a smoothly bounded convex domain $\Omega\subset {\mathbb{R}}^2$, where $m>1$, $\alpha>0, \beta>0$ and $D>0$ are parameters, the response function $f$ satisfies $f\in C^1([0,\infty)), f(0)=0, f(w)>0$ in $(0,\infty)$. This system describes the density-suppressed motility of Eeshcrichia coli cells in process of spatio-temporal pattern formation via so-called self-trapping mechanisms. Based on the duality argument, it is shown that for suitable large $D$ the problem admits at least one global weak solution $(u,v,w)$ which will asymptotically converge to the spatially uniform equilibrium $(\overline{u_0}+\beta \overline{w_0},\overline{u_0}+\beta \overline{w_0},0)$ with $\overline{u_0}=\frac1{|\Omega|}\int_{\Omega}u(x,0)dx $ and $\overline{w_0}=\frac1{|\Omega|}\int_{\Omega}w(x,0)dx $ in $L^\infty(\Omega)$.