Global classical solutions for a class of reaction-diffusion system with density-suppressed motility

Abstract

This paper is concerned with a class of reaction-diffusion system with density-suppressed motility \begin{equation*} \begin{cases} u_{t}=\Delta(\gamma(v) u)+\alpha u F(w), & x \in \Omega, \quad t>0, \\ v_{t}=D \Delta v+u-v, & x \in \Omega, \quad t>0, \\ w_{t}=\Delta w-u F(w), & x \in \Omega, \quad t>0, %\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=\frac{\partial w}{\partial \nu}=0, & x \in \partial \Omega, \quad t>0, \\ %(u, v, w)(x, 0)=\left(u_{0}, v_{0}, w_{0}\right)(x), & x \in \Omega, \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^n~(n\leq 2)$, where $\alpha>0$ and $D>0$ are constants. The random motility function $\gamma$ satisfies \begin{equation*} \gamma\in C^3((0,+\infty)),\ \gamma>0,\ \gamma'<0\,\ \text{on}\,\ (0,+\infty) \ \ \text{and}\ \ \lim_{v\rightarrow+\infty}\gamma(v)=0. \end{equation*} %and %\begin{equation*} %\lim_{x\rightarrow+\infty}\gamma(x)=0. %\end{equation*} The intake rate function $F$ satisfies \begin{equation*} F\in C^1([0,+\infty)),\,F(0)=0\,\ \text{and}\ \,F>0\,\ \text{on}\,\ (0,+\infty). \end{equation*} We show that the above system admits a unique global classical solution for all non-negative initial data $$ u_0\in C^0(\overline{\Omega}),\,v_0\in W^{1,\infty}(\Omega),\,w_0\in W^{1,\infty}(\Omega). $$ Moreover, if there exist $k>0$ and $\overline{v}>0$ such that \begin{equation*} \inf_{v>\overline{v}}v^k\gamma(v)>0, \end{equation*} then the global solution is bounded uniformly in time.