Boundedness, Stabilization, and Pattern Formation Driven by Density-Suppressed Motility

We are concerned with the following density-suppressed motility model: uₜ=Δ (γ(v) u)+μ u(1-u); vₜ=Δ v+ u-v, in a bounded smooth domain Ω⊂ R² with homogeneous Neumann boundary conditions, where the motility function γ(v)∈ 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.