global existence

reaction diffusion

wave equation

chemotaxis model

Calculus of variations and partial differential equations

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.

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