BOUNDEDNESS, STABILIZATION, AND PATTERN FORMATION DRIVEN BY DENSITY-SUPPRESSED MOTILITY

Abstract

We are concerned with the following density-suppressed motility model: u(t) = Delta(gamma(v)u) + mu u(1 - u); v(t) = Delta v + u - v, in a bounded smooth domain Omega subset of R-2 with homogeneous Neumann boundary conditions, where the motility function gamma(v) is an element of C-3([0, infinity)), gamma(v) > 0, gamma'(v) < 0 for all v >= 0, lim(v ->infinity) gamma(v) = 0, and lim(v ->infinity) gamma'(v)/gamma(v) exists. The model is proposed to advocate a new possible mechanism: density-suppressed motility can induce spatio-temporal pattern formation through self-trapping. The major technical difficulty in the analysis of above density-suppressed motility model is the possible degeneracy of diffusion from the condition lim(v ->infinity) gamma(v) = 0. In this paper, by treating the motility function gamma(v) as a weight function and employing the method of weighted energy estimates, we derive the a priori L-infinity-bound of v to rule out the degeneracy and establish the global existence of classical solutions of the above problem with a uniform-in-time bound. Furthermore, we show if it mu > K-0/16 with K-0 = max(0 <= v <=infinity) vertical bar gamma'(v)vertical bar(2)/gamma(v), the constant steady state (1,1) is globally asymptotically stable and, hence, pattern formation does not exist. For small mu > 0, we perform numerical simulations to illustrate aggregation patterns and wave propagation formed by the model.