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

Source 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 $\Omega\subset \mathbb{R}^2$ with homogeneous Neumann boundary conditions, where the motility function $\gamma(v)\in C^3([0,\infty))$, $\gamma (v)>0$, $\gamma'(v)<0$ for all $v\geq 0$, $\lim_{v \to \infty}\gamma(v)=0$, and $\lim_{v \to \infty}\frac{\gamma'(v)}{\gamma(v)}$ 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 $\lim_{v \to \infty}\gamma(v)=0$. 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 $L^\infty$-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 $\mu>\frac{K_0}{16}$ with $K_0=\max_{0\leq v \leq \infty}\frac{|\gamma'(v)|^2}{\gamma(v)}$, 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.