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

TL;DR

This paper analyzes a reaction-diffusion system with density-dependent motility, proving global boundedness of solutions and describing their long-term asymptotic behavior depending on parameter .

Contribution

It establishes the existence, uniqueness, and boundedness of solutions, and characterizes their asymptotic states for different parameter regimes.

Findings

Solutions are globally bounded and unique.

Solutions converge to specific steady states over time.

Behavior depends on the parameter ; either decay to zero or stabilize to a positive state.

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=\Delta(\gamma(v)u)+\alpha u F(w) -\theta u, &x\in \Omega, ~~t>0,\\ v_t=D\Delta v+u-v,& x\in \Omega, ~~t>0,\\ w_t=\Delta w-uF(w),& x\in \Omega, ~~t>0, \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}= \frac{\partial w}{\partial \nu}=0,&x\in \partial\Omega, ~~t>0,\\ (u,v,w)(x,0)=(u_0,v_0,w_0)(x), & x\in\Omega, \end{cases} \end{equation*} in a bounded domain $\Omega\subset\R^2$ 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 $\gamma(v)\in C^{3}([0,\infty)),~0<\gamma_{1}\leq\gamma(v)\leq \gamma_2, \ |\gamma'(v)|\leq \eta$ 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 $\gamma_1, \gamma_2$ 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 $(0,0,w_*)$ in $L^\infty$ with some $w_*>0$ as time tends to infinity, while if $\theta=0$, the solution $(u,v,w)$ will asymptotically converge to $(u_*,u_*,0)$ in $L^\infty$ with $u_*=\frac{1}{|\Omega|}(\|u_0\|_{L^1}+\alpha\|w_0\|_{L^1})$ if $D>0$ is suitably large.