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 (0.1){ut=Δ(γ(v)u)+μu(1−u)−Δv+v=u in a smooth bounded domain Ω⊂Rn, n≥1 with no-flux boundary conditions. Here, μ≥0 is any given constant. The function γ(⋅) represents a signal-dependent diffusion motility and is decreasing in v which models a density-suppressed motility in the process of stripe pattern formation through the self-trapping mechanism [9,20]. The major difficulty in analysis lies in the possible degeneracy of diffusion as v↗+∞. 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 functions satisfying γ(v)>0 and γ′(v)≤0 for v≥0. Then we prove global existence of classical solution for (0.1) in the two-dimensional setting with any μ≥0. Moreover, the global solution is proven to be uniform-in-time bounded if either 1/γ satisfies certain polynomial growth condition or μ>0. Besides, we pay particular attention to the specific case γ(v)=e−v with μ=0. Under the circumstances, system (0.1) becomes of great interest because it shares the same set of equilibria as well as the Lyapunov functional with the classical Keller–Segel model. A novel critical phenomenon in the two-dimensional setting is observed that with any initial datum of sub-critical mass, the global solution is proved to be uniform-in-time bounded, while with certain initial datum of super-critical mass, the global solution will become unbounded as time goes to infinity. Namely, blowup takes place in infinite time rather than finite time in our model which is distinct from the well-known fact that certain initial data of super-critical mass will enforce a finite-time blowup for the classical Keller–Segel system.