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

This paper is concerned with a class of reaction-diffusion system with density-suppressed motility

$$\{\begin{array}{ll}{u}_t=\mathrm{\Delta }(\gamma (v)u)+\alpha uF(w),& x\in \mathrm{\Omega },t>0,\\ v_t=D\mathrm{\Delta }v+u-v,& x\in \mathrm{\Omega },t>0,\\ w_t=\mathrm{\Delta }w-uF(w),& x\in \mathrm{\Omega },t>0,\end{array}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\le 2)$, where $\alpha >0$ and $D>0$ are constants. The random motility function $\gamma$ satisfies

$$\gamma \in C^3((0,+\mathrm{\infty })),\gamma >0,{\gamma }^{\prime }<0\text{on}(0,+\mathrm{\infty })\text{and}\underset{v\to +\mathrm{\infty }}{lim}\gamma (v)=0.$$

The intake rate function $F$ satisfies $F\in C^1([0,+\mathrm{\infty })),F(0)=0\text{and}F>0\text{on}(0,+\mathrm{\infty })$. We show that the above system admits a unique global classical solution for all non-negative initial data $u_0\in W^{1,\mathrm{\infty }}(\mathrm{\Omega }),v_0\in W^{1,\mathrm{\infty }}(\mathrm{\Omega }),w_0\in W^{1,\mathrm{\infty }}(\mathrm{\Omega })$. Moreover, if there exist $k>0$ and $\overline{v}>0$ such that

$$\underset{v>\overline{v}}{inf}{v}^k\gamma (v)>0,$$

then the global solution is bounded uniformly in time.