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

Abstract

<abstract><p>This paper is concerned with a class of reaction-diffusion system with density-suppressed motility</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} u_{t} = \Delta(\gamma(v) u)+\alpha u F(w), &amp; x \in \Omega, \quad t&gt;0, \\ v_{t} = D \Delta v+u-v, &amp; x \in \Omega, \quad t&gt;0, \\ w_{t} = \Delta w-u F(w), &amp; x \in \Omega, \quad t&gt;0, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>under homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega\subset \mathbb{R}^n\; (n\leq 2) $, where $ \alpha &gt; 0 $ and $ D &gt; 0 $ are constants. The random motility function $ \gamma $ satisfies</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \gamma\in C^3((0, +\infty)), \ \gamma&gt;0, \ \gamma'&lt;0\, \ \text{on}\, \ (0, +\infty) \ \ \text{and}\ \ \lim\limits_{v\rightarrow +\infty}\gamma(v) = 0. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>The intake rate function $ F $ satisfies $ F\in C^1([0, +\infty)), \, F(0) = 0\, \ \text{and}\ \, F &gt; 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 &gt; 0 $ and $ \overline{v} &gt; 0 $ such that</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{equation*} \inf\limits_{v&gt;\overline{v}}v^k\gamma(v)&gt;0, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>then the global solution is bounded uniformly in time.</p></abstract>