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.
Categories: math.AP