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

TL;DR

This paper proves the global existence of classical solutions for a kinetic model of pattern formation with density-suppressed motilities, addressing degeneracy issues and revealing a critical blowup phenomenon in two dimensions.

Contribution

The authors develop a new method to prevent finite-time degeneracy and establish global solutions in 2D for a class of nonlinear diffusion models with density-dependent motility functions.

Findings

Global existence of classical solutions in 2D for all .

Solutions are uniformly bounded under certain conditions.

Blowup occurs in infinite time for specific parameters.

Abstract

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.