Abstract

The degenerate Keller-Segel type system

$$\{\begin{array}{cccc}{\partial }_{t}u& =& \nabla ·(u^{m-1}\nabla u)-\nabla ·(u\nabla v),& x\in \Omega ,t>0,\\ 0& =& \Delta v-\mu +u,\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}{\int }_{\Omega }v=0,\hspace{1em}\mu =\frac{1}{\left|\Omega \right|}{\int }_{\Omega }u,\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}& x\in \Omega ,t>0,\end{array}$$

is considered in balls $\Omega =B_R(0)\subset {ℝ}^n$ with $n\ge 1,$ R > 0 and m > 1. Our main results reveal that throughout the entire degeneracy range $m\in (1,\infty ),$ the interplay between degenerate diffusion and cross-diffusive attraction herein can enforce persistent localization of solutions inside a compact subset of Ω, no matter whether solutions remain bounded or blow up. More precisely, it is shown that for arbitrary $\mu >0,\sigma \in (0,1)$ and $\theta \in (0,\sigma )$ one can find $R_{\mathrm{\star }}=R_{\mathrm{\star }}(n,m,\mu ,\sigma ,\theta )>0$ such that if $R\ge R_{\mathrm{\star }}$ and $u_0\in L^{\infty }(\Omega )$ is nonnegative and radially symmetric with $\frac{1}{\left|\Omega \right|}{\int }_{\Omega }{u}_0=\mu$ and

$$\frac{1}{|B_r(0)|}{\int }_{B_r(0)}{u}_0\ge \frac{\mu }{{\theta }^n}\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\text{for}\mathrm{ }\text{all}\mathrm{ }r\in (0,\theta R),$$

then a corresponding zero-flux type initial-boundary value problem admits a radial weak solution (u, v), extensible up to a maximal time $T_{\mathit{max}}\in (0,\infty ]$ and satisfying $\underset{t\nearrow T_{\mathit{max}}}{\lim }\Vert u(·,t){\Vert }_{L^{\infty }(\Omega )}=\infty$ if $T_{\mathit{max}}<\infty ,$ which has the additional property that

$$\text{supp}u(·,t)\subset {\overline{B}}_{\sigma R}(0)\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\text{for}\mathrm{ }\text{all}\mathrm{ }t\in (0,T_{\mathit{max}}).$$

In particular, this conclusion is seen to be valid whenever u₀ is radially nonincreasing with $\text{supp}{u}_0\subset {\overline{B}}_{\theta R}(0).$

摘要

退化的 Keller-Segel 型系统

$$\{\begin{array}{cccc}{\partial }_{吨}你& =& \nabla ·({你}^{米-\mathrm{1个}}\nabla 你)-\nabla ·(你\nabla v),& X\in \mathrm{欧姆},吨>0,\\ 0& =& △v-\mu +你,\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}{\int }_{\mathrm{欧姆}}v=0,\hspace{1em}\mu =\frac{\mathrm{1个}}{\left|\mathrm{欧姆}\right|}{\int }_{\mathrm{欧姆}}你,\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}& X\in \mathrm{欧姆},吨>0,\end{array}$$

被认为是球$\mathrm{欧姆}={乙}_R(0)\subset {ℝ}^n$和$n\ge \mathrm{1个},$ R  > 0 和m  > 1。我们的主要结果表明，在整个简并范围内$米\in (\mathrm{1个},\infty ),$这里退化扩散和交叉扩散吸引之间的相互作用可以强制解决方案在 Ω 的紧凑子集中持续定位，无论解决方案是否保持有界或爆炸。更准确地说，它表明对于任意$\mu >0,\sigma \in (0,\mathrm{1个})$和$\theta \in (0,\sigma )$可以找到$R_{\mathrm{\star }}=R_{\mathrm{\star }}(n,米,\mu ,\sigma ,\theta )>0$这样如果$R\ge R_{\mathrm{\star }}$和${你}_0\in {\mathrm{大号}}^{\infty }(\mathrm{欧姆})$是非负的并且径向对称于$\frac{\mathrm{1个}}{\left|\mathrm{欧姆}\right|}{\int }_{\mathrm{欧姆}}{你}_0=\mu$和

$$\frac{\mathrm{1个}}{|{乙}_r(0)|}{\int }_{{乙}_r(0)}{你}_0\ge \frac{\mu }{{\theta }^n}\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\text{为了}\mathrm{ }\text{全部}\mathrm{ }r\in (0,\theta R),$$

那么相应的零通量类型初始边界值问题允许径向弱解 ( u , v )，可扩展到最大时间${吨}_{\mathit{最大限度}}\in (0,\infty ]$和满足$\underset{吨\nearrow {吨}_{\mathit{最大限度}}}{\mathrm{林}}\Vert 你(·,吨){\Vert }_{{\mathrm{大号}}^{\infty }(\mathrm{欧姆})}=\infty$如果${吨}_{\mathit{最大限度}}<\infty ,$它具有额外的属性

$$\text{支持}你(·,吨)\subset {\overline{乙}}_{\sigma R}(0)\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\text{为了}\mathrm{ }\text{全部}\mathrm{ }吨\in (0,{吨}_{\mathit{最大限度}}).$$

特别地，只要u ₀径向不增加，这个结论就被认为是有效的$\text{支持}{你}_0\subset {\overline{乙}}_{\theta R}(0).$