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

다국어 초록 (Multilingual Abstract)

<P>We are concerned with the following density-suppressed motility model: <TEX>$u_t=\Delta (\gamma(v) u)+\mu u(1-u); v_t=\Delta v+ u-v,$</TEX> in a bounded smooth domain <TEX>$\Omega\subset \mathbb{R}^2$</TEX> with homogeneous Neumann boundary conditions, where the motility function <TEX>$\gamma(v)\in C^3([0,\infty))$</TEX>, <TEX>$\gamma (v)>0$</TEX>, <TEX>$\gamma'(v)<0$</TEX> for all <TEX>$v\geq 0$</TEX>, <TEX>$\lim_{v \to \infty}\gamma(v)=0$</TEX>, and <TEX>$\lim_{v \to \infty}\frac{\gamma'(v)}{\gamma(v)}$</TEX> 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 <TEX>$\lim_{v \to \infty}\gamma(v)=0$</TEX>. In this paper, by treating the motility function <TEX>$\gamma(v)$</TEX> as a weight function and employing the method of weighted energy estimates, we derive the a priori <TEX>$L^\infty$</TEX>-bound of <TEX>$v$</TEX> 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 <TEX>$\mu>\frac{K_0}{16}$</TEX> with <TEX>$K_0=\max_{0\leq v \leq \infty}\frac{|\gamma'(v)|^2}{\gamma(v)}$</TEX>, the constant steady state (1,1) is globally asymptotically stable and, hence, pattern formation does not exist. For small <TEX>$\mu>0$</TEX>, we perform numerical simulations to illustrate aggregation patterns and wave propagation formed by the model.</P>