凯勒-西格尔系统中无限时间炸裂的存在性和稳定性

IF 2.6 1区 数学 Q1 MATHEMATICS, APPLIED Archive for Rational Mechanics and Analysis Pub Date : 2024-06-18 DOI:10.1007/s00205-024-02006-7
Juan Dávila, Manuel del Pino, Jean Dolbeault, Monica Musso, Juncheng Wei
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引用次数: 0

摘要

我们考虑临界质量情况(int _{\mathbb {R}}^2} u_0(x)\, \textrm{d}x = 8\pi \),它对应于有限时间膨胀和自相似扩散趋零之间的精确临界点。我们找到一个质量为\(8\pi \)的径向函数\(u_0^*\),对于任何足够接近\(u_0^*\)的初始条件和质量为\(8\pi \)的初始条件,(\(*\))的解u(x, t)是全局定义的,并且在无限时间内炸毁。由于(t\rightarrow +\infty \)它有近似的轮廓 $$\begin{aligned} u(x,t) \approx \frac{1}\{lambda ^2(t)} U\left( \frac{x-\xi (t)}\{lambda (t)} \right) 、\quad U(y)= \frac{8}{(1+|y|^2)^2}, \end{aligned}$$where \(\lambda (t) \approx \frac{c}{\sqrt\{log t}}\), \(\xi (t)\rightarrow q\) for some \(c>;0) and\(q\in {\mathbb {R}}^2\).这一结果肯定地回答了 Ghoul 和 Masmoudi(Commun Pure Appl Math 71:1957-2015, 2018)中提出的非径向稳定性猜想。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System

Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system

We consider the critical mass case \(\int _{{\mathbb {R}}^2} u_0(x)\, \textrm{d}x = 8\pi \), which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function \(u_0^*\) with mass \(8\pi \) such that for any initial condition \(u_0\) sufficiently close to \(u_0^*\) and mass \(8\pi \), the solution u(xt) of (\(*\)) is globally defined and blows-up in infinite time. As \(t\rightarrow +\infty \) it has the approximate profile

$$\begin{aligned} u(x,t) \approx \frac{1}{\lambda ^2(t)} U\left( \frac{x-\xi (t)}{\lambda (t)} \right) , \quad U(y)= \frac{8}{(1+|y|^2)^2}, \end{aligned}$$

where \(\lambda (t) \approx \frac{c}{\sqrt{\log t}}\), \(\xi (t)\rightarrow q\) for some \(c>0\) and \(q\in {\mathbb {R}}^2\). This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).

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来源期刊
CiteScore
5.10
自引率
8.00%
发文量
98
审稿时长
4-8 weeks
期刊介绍: The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.
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