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{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, \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\limits_{v\rightarrow +\infty}\gamma(v) = 0. \end{equation*} $

The intake rate function $ F $ satisfies $ F\in C^1([0, +\infty)), \, F(0) = 0\, \ \text{and}\ \, F > 0\, \ \text{on}\, \ (0, +\infty) $. We show that the above system admits a unique global classical solution for all non-negative initial data $ u_0\in W^{1, \infty}(\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\limits_{v>\overline{v}}v^k\gamma(v)>0, \end{equation*} $

then the global solution is bounded uniformly in time.