密度抑制动力学斑图形成动力学模型的整体存在性
Global Existence for a Kinetic Model of Pattern Formation with Density-suppressed Motilities
摘要：
本文考虑了光滑有界区域$Ω\subset\mathbb{R}^n$中的斑图形成动力学模型\begin{equation} \begin{cases} u_t=Δ(γ(v)u)+μu(1-u) -Δv+v=u \end{cases} \qquad (0.1) \end{equation}，无通量边界条件下的$n\geq1$的整体经典解的存在性.这里，$μ\geq0$是任何给定的常数。函数$γ(\cdot)$代表信号依赖性扩散运动性，并在$v$中降低，$v$通过自陷机制模拟条纹图案形成过程中的密度抑制运动性[8，20]。分析中的主要困难在于扩散可能退化为$v\nearrow+\infty.$在目前的贡献，基于一个微妙的观察的非线性结构，我们开发了一个新的方法，以排除有限时间退化在任何空间维的所有光滑运动函数满足$γ(v)>0$和$γ'(v)\leq0$为$v\geq0$。在二维情形下，证明了方程（0.1）的经典解的整体存在性，并给出了任意$μ\geq0$ .此外，如果$1/γ$满足一定的多项式增长条件或$μ>0.$，我们将特别注意的情况下，$γ(v)=e^{-v}$与$μ=0$的整体解被证明是一致的时间有界。在我们的模型中，在无限时间而不是有限时间内发生爆破的二维环境中观察到一种新的临界现象。
英文摘要：
In this paper, we consider global existence of classical solutions to the following kinetic model of pattern formation \begin{equation} \begin{cases} u_t=Δ(γ(v)u)+μu(1-u) -Δv+v=u \end{cases} \qquad (0.1) \end{equation}in a smooth bounded domain $Ω\subset\mathbb{R}^n$, $n\geq1$ with no-flux boundary conditions. Here, $μ\geq0$ is any given constant. The function $γ(\cdot)$ represents a signal-dependent diffusion motility and is decreasing in $v$ which models a density-suppressed motility in process of stripe pattern formation through self-trapping mechanism [8,20]. The major difficulty in analysis lies in the possible degeneracy of diffusion as $v\nearrow+\infty.$ In the present contribution, based on a subtle observation of the nonlinear structure, we develop a new method to rule out finite-time degeneracy in any spatial dimension for all smooth motility function satisfying $γ(v)>0$ and $γ'(v)\leq0$ for $v\geq0$. Then we prove global existence of classical solution for (0.1) in the two-dimensional setting with any $μ\geq0$. Moreover, the global solution is proven to be uniform-in-time bounded if either $1/γ$ satisfies certain polynomial growth condition or $μ>0.$ Besides, we pay particular attention to the specific case $γ(v)=e^{-v}$ with $μ=0$. A novel critical phenomenon in the two-dimensional setting is observed where blowup takes place in infinite time rather than finite time in our model.
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