论文标题
具有密度抑制运动的模式形成动力学模型的全球存在
Global Existence for a Kinetic Model of Pattern Formation with Density-suppressed Motilities
论文作者
论文摘要
在本文中，我们将经典解决方案考虑到以下模式的全球存在\ Begin \ begin {equation} \ begin {cases} u_t =δ（γ（v）u）+μu（1-u）+μ（1-U）-ΔV+v = u \ end end {cases} {cases} $ω\ subset \ mathbb {r}^n $，$ n \ geq1 $，具有无通用边界条件。在这里，$μ\ geq0 $是给定常数。函数$γ（\ cdot）$代表信号依赖性扩散运动性，并且在$ v $中降低，该$ V $在条纹模式形成过程中通过自捕获机制模拟了密度抑制的运动性[8,20]。分析的主要困难在于，扩散的可能退化为$ v \近+\ iftty。$在本捐款中，基于对非线性结构的细微观察，我们开发了一种新方法来排除所有空间变性的有限时间退化，以满足所有平稳的动力函数，以满足$γ$ 0 $ 0 $ $ $ up $ up $ up $γ'（v）$γ'（v）$γ'（v）$γ'（v）$γ'（v）。然后，我们证明了在二维设置中使用任何$μ\ geq0 $的二维设置中的经典解决方案。此外，如果$ 1/γ$满足某些多项式生长条件或$μ> 0，则全球解决方案被证明是统一的界限。观察到在二维环境中的一种新颖的关键现象，在无限时间而不是在我们的模型中进行有限的时间进行爆炸。
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.
