Comparison methods for a Keller–Segel-type model of pattern formations with density-suppressed motilities

Abstract

This paper is concerned with global existence of classical solutions as well as occurrence of infinite-time blowups to the following fully parabolic system

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\varDelta (\gamma (v)u)\\ v_t-\varDelta v+v=u \end{array}\right. } \end{aligned}$$

(1)

in a smooth bounded domain $\varOmega \subset {\mathbb {R}}^n$, $n\ge 1$ with no-flux boundary conditions. This model was recently proposed in Fu et al. (Phys Rev Lett 108:198102, 2012) and Liu et al. (Science 334:238, 2011) 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. An essential difficulty in analysis comes from the possible degeneracy as $v\nearrow \infty .$ In this work we develop a novel comparison method to tackle the degeneracy issue, which greatly differs from the conventional energy method in literature. An explicit point-wise upper-bound estimate for v is obtained for the first time, which shows that v(x, t) grows point-wisely at most exponentially in time. An intrinsic mechanism is then unveiled that the finite-time degeneracy is prohibited in any spatial dimension with a generic decreasing $\gamma$. With such new findings, we further study global existence of classical solutions when $n\le 3$ and discuss uniform-in-time boundedness when $\gamma (\cdot )$ decreases algebraically at large signal concentrations. Besides, a new critical-mass phenomenon in dimension two is observed if $\gamma (v)=e^{-v}$. Indeed, we prove that the classical solution always exists globally and remains uniformly-in-time bounded in the sub-critical case, while in the super-critical case a blowup may take place in infinite time rather than finite time.