Global Existence for a Kinetic Model of Pattern Formation with Density-suppressed Motilities

In this paper, we consider global existence of classical solutions to the following kinetic model of pattern formation \begin{equation} \begin{cases} u_t=\Delta (\gamma (v)u)+\mu u(1-u) -\Delta v+v=u \end{cases} \qquad (0.1) \end{equation}in a smooth bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq1$ with no-flux boundary conditions. Here, $\mu\geq0$ is any given constant. The function $\gamma(\cdot)$ represents a signal-dependent diffusion motility and is decreasing in $v$ which models a density-suppressed motility in process of stripe pattern formation through self-trapping mechanism [8,20]. The major difficulty in analysis lies in the possible degeneracy of diffusion as $v\nearrow+\infty.$ In the present contribution, based on a subtle observation of the nonlinear structure, we develop a new method to rule out finite-time degeneracy in any spatial dimension for all smooth motility function satisfying $\gamma(v)>0$ and $\gamma'(v)\leq0$ for $v\geq0$. Then we prove global existence of classical solution for (0.1) in the two-dimensional setting with any $\mu\geq0$. Moreover, the global solution is proven to be uniform-in-time bounded if either $1/\gamma$ satisfies certain polynomial growth condition or $\mu>0.$ Besides, we pay particular attention to the specific case $\gamma(v)=e^{-v}$ with $\mu=0$. A novel critical phenomenon in the two-dimensional setting is observed where blowup takes place in infinite time rather than finite time in our model.