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

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 $Ω\subset\R^2$ with smooth boundary, $α$ and $θ$ are non-negative constants and $ν$ denotes the outward normal vector of $\partial Ω$. The random motility function $γ(v)$ and functional response function $F(w)$ satisfy the following assumptions: \begin{itemize} \item $γ(v)\in C^{3}([0,\infty)),~0<γ_{1}\leqγ(v)\leq γ_2, \ |γ'(v)|\leq η$ for all $v\geq0$; \item $F(w)\in C^1([0,\infty)), F(0)=0,F(w)>0 \ \mathrm{in}~(0,\infty)~\mathrm{and}~F'(w)>0 \ \mathrm{on}\ \ [0,\infty)$ \end{itemize} for some positive constants $γ_1, γ_2$ and $η$. 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 $θ>0$, the solution $(u,v,w)$ will converge to $(0,0,w_*)$ in $L^\infty$ with some $w_*>0$ as time tends to infinity, while if $θ=0$, the solution $(u,v,w)$ will asymptotically converge to $(u_*,u_*,0)$ in $L^\infty$ with $u_*=\frac{1}{|Ω|}(\|u_0\|_{L^1}+α\|w_0\|_{L^1})$ if $D>0$ is suitably large.

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