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

Abstract

This paper is concerned with the density-suppressed motility model: \(u_{t}=\Delta \left( \displaystyle \frac{u^m}{v^\alpha }\right) +\beta uf(w), v_{t}=D\Delta v-v+u, w_{t}=\Delta w-uf(w)\) 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 the 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 \(L^\infty (\Omega )\).