Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility

Abstract

We consider the initial-boundary value problem of a system of reaction-diffusion equations with density-dependent motility \begin{equation*}\label{e1}\tag{$\ast$} \begin{cases} u_t=Δ(γ(v)u)+αu F(w) -θu, &x\in Ω, ~~t>0,\\ v_t=DΔv+u-v,& x\in Ω, ~~t>0,\\ w_t=Δw-uF(w),& x\in Ω, ~~t>0, \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}= \frac{\partial w}{\partial ν}=0,&x\in \partialΩ, ~~t>0,\\ (u,v,w)(x,0)=(u_0,v_0,w_0)(x), & x\inΩ, \end{cases} \end{equation*} in a bounded domain $\Omega \subset {\mathbb{R}}^2$ with smooth boundary, $\alpha$ and $\theta$ are non-negative constants and $\nu$ denotes the outward normal vector of $\partial \Omega$. The random motility function $\gamma (v)$ and functional response function $F(w)$ satisfy the following assumptions: \begin{itemize} \item $\gamma (v)\in C^3([0,\infty )),0<{\gamma }_1\le \gamma (v)\le {\gamma }_2,|{\gamma }^{\prime }(v)|\le \eta$ for all $v\ge 0$; \item $F(w)\in C^1([0,\infty )),F(0)=0,F(w)>0\mathrm{in}(0,\infty )\mathrm{and}{F}^{\prime }(w)>0\mathrm{on}[0,\infty )$ \end{itemize} for some positive constants ${\gamma }_1,{\gamma }_2$ and $\eta$. Based on the method of weighted energy estimates and Moser iteration, we prove that the problem \eqref{e1} has a unique classical global solution uniformly bounded in time. Furthermore we show that if $\theta >0$, the solution $(u,v,w)$ will converge to $(0,0,w_{\ast })$ in $L^{\infty }$ with some $w_{\ast }>0$ as time tends to infinity, while if $\theta =0$, the solution $(u,v,w)$ will asymptotically converge to $(u_{\ast },u_{\ast },0)$ in $L^{\infty }$ with $u_{\ast }=\frac{1}{|\Omega |}(\Vert u_0{\Vert }_{L^1}+\alpha \Vert w_0{\Vert }_{L^1})$ if $D>0$ is suitably large.