Comparison Methods for a Keller--Segel-type Model of Pattern Formations with Density-suppressed Motilities

This paper is concerned with global existence as well as infinite-time blowups of classical solutions to the following fully parabolic kinetic system \begin{equation} \begin{cases} u_t=\Delta (\gamma (v)u) v_t-\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. This model was recently proposed in [8,20] to describe the process of stripe pattern formations via the so-called self-trapping mechanism. The system features a signal-dependent motility function $\gamma(\cdot)$, which is decreasing in $v$ and will vanish as $v$ tends to infinity. The major difficulty in analysis comes from the possible degeneracy as $v\nearrow+\infty.$ In this work we develop a new comparison method different from the conventional energy method in literature, which reveals a striking fact that there is no finite-time degeneracy in this system. More precisely, we use comparison principles for elliptic and parabolic equations to prove that degeneracy cannot take place in finite time in any spatial dimensions for all smooth motility functions satisfying $\gamma(s)>0$, $\gamma'(s)\leq0$ when $s\geq0$ and $\lim\limits_{s\rightarrow+\infty}\gamma(s)=0.$ Then we investigate global existence of classical solutions to (0.1) when $n\leq3$ and discuss the uniform-in-time boundedness under certain growth conditions on $1/\gamma.$ In particular, we consider system (0.1) with $\gamma(v)=e^{-v}$. We prove that classical solution always exists globally, which must be uniformly-in-time bounded with arbitrary initial data of sub-critical mass. On the contrary, with certain initial data of super-critical mass, the solution will become unbounded at time infinity which differs from the finite-time blowup behavior of the Keller--Segel model.