Boundedness, stabilization, and pattern formation driven by density-suppressed motility

Abstract

We are concerned with the following density-suppressed motility model: Ut = Δ(γ(v)u) + μu(1- u); vt = Δv + u- v, in a bounded smooth domain Ω ⊆ R2 with homogeneous Neumann boundary conditions, where the motility function γ(v) ϵ C3([0,∞)), γ(v) > 0, γ(v) < 0 for all v ≥ 0, limv→∞γ(v) = 0, and limv→∞ γ (v) γ(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 limv→∞ γ(v) = 0. In this paper, by treating the motility function γ(v) as a weight function and employing the method of weighted energy estimates, we derive the a priori L∞-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 μ > K0 16 with K0 = max0≤v≤∞ γ (v)2 γ(v) , the constant steady state (1, 1) is globally asymptotically stable and, hence, pattern formation does not exist. For small μ > 0, we perform numerical simulations to illustrate aggregation patterns and wave propagation formed by the model.