论文标题
凯勒（Keller）的比较方法 - 具有密度抑制运动的模式形成模型
Comparison Methods for a Keller--Segel-type Model of Pattern Formations with Density-suppressed Motilities
论文作者
论文摘要
本文与以下完全抛物线的动力学系统\ begin \ begin {equation} \ begin {cases} u_t =δ（γ（v _t-ΔV+v = us） $ω\ subset \ mathbb {r}^n $，$ n \ geq1 $，具有无通用边界条件。最近在[8,20]中提出了该模型，以通过所谓的自捕捞机制来描述条纹模式形成的过程。该系统具有信号依赖的运动函数$γ（\ cdot）$，该$在$ v $中降低，并且随着$ v $的趋势而消失。分析的主要困难来自可能的退化，因为$ v \近+\ infty。$在这项工作中，我们开发了一种与文献中常规能量方法不同的新比较方法，这表明该系统中没有有限的时间退化。更确切地说，我们使用椭圆形和抛物线方程的比较原理证明，在任何空间维度上都不会在有限的时间内进行脱位，以满足满足$γ（s）> 0 $，$γ'（s）\ leq0 $时的所有平滑运动功能$ \ lim \ limits_ {s \ rightarrow+\ infty}γ（s）= 0。$然后，当$ n \ leq3 $ $ n \ leq3 $时，我们调查了经典解决方案的全球存在，并在某些$ 1/γ的某些增长条件下讨论了$ 1/γ的均匀界限。我们证明，经典解决方案始终存在于全球，必须在次临界质量的任意初始数据中统一地界定。相反，使用超临界质量的某些初始数据，该解决方案将在时间无限属性上变得无限，这与凯勒（Keller）模型的有限时间爆炸行为不同。
This paper is concerned with global existence as well as infinite-time blowups of classical solutions to the following fully parabolic kinetic system \begin{equation} \begin{cases} u_t=Δ(γ(v)u) v_t-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. This model was recently proposed in [8,20] to describe the process of stripe pattern formations via the so-called self-trapping mechanism. The system features a signal-dependent motility function $γ(\cdot)$, which is decreasing in $v$ and will vanish as $v$ tends to infinity. The major difficulty in analysis comes from the possible degeneracy as $v\nearrow+\infty.$ In this work we develop a new comparison method different from the conventional energy method in literature, which reveals a striking fact that there is no finite-time degeneracy in this system. More precisely, we use comparison principles for elliptic and parabolic equations to prove that degeneracy cannot take place in finite time in any spatial dimensions for all smooth motility functions satisfying $γ(s)>0$, $γ'(s)\leq0$ when $s\geq0$ and $\lim\limits_{s\rightarrow+\infty}γ(s)=0.$ Then we investigate global existence of classical solutions to (0.1) when $n\leq3$ and discuss the uniform-in-time boundedness under certain growth conditions on $1/γ.$ In particular, we consider system (0.1) with $γ(v)=e^{-v}$. We prove that classical solution always exists globally, which must be uniformly-in-time bounded with arbitrary initial data of sub-critical mass. On the contrary, with certain initial data of super-critical mass, the solution will become unbounded at time infinity which differs from the finite-time blowup behavior of the Keller--Segel model.
