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

Abstract

This paper is concerned with the density-suppressed motility model: $u_t=\Delta (\frac{u^m}{v^{\alpha }})+\beta uf(w),v_t=D\Delta v-v+u,w_t=\Delta w-uf(w)$ 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 $\overline{u_0}=\frac{1}{|\Omega |}{\int }_{\Omega }u(x,0)dx$ and $\overline{w_0}=\frac{1}{|\Omega |}{\int }_{\Omega }w(x,0)dx$ in $L^{\infty }(\Omega )$.

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