  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
$L^\infty(\Omega)$.

Analysis of PDEs