Juan Dávila, Manuel del Pino, Jean Dolbeault, Monica Musso, Juncheng Wei
{"title":"凯勒-西格尔系统中无限时间炸裂的存在性和稳定性","authors":"Juan Dávila, Manuel del Pino, Jean Dolbeault, Monica Musso, Juncheng Wei","doi":"10.1007/s00205-024-02006-7","DOIUrl":null,"url":null,"abstract":"<div><p>Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system </p><div><figure><div><div><picture><img></picture></div></div></figure></div><p> We consider the critical mass case <span>\\(\\int _{{\\mathbb {R}}^2} u_0(x)\\, \\textrm{d}x = 8\\pi \\)</span>, which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function <span>\\(u_0^*\\)</span> with mass <span>\\(8\\pi \\)</span> such that for any initial condition <span>\\(u_0\\)</span> sufficiently close to <span>\\(u_0^*\\)</span> and mass <span>\\(8\\pi \\)</span>, the solution <i>u</i>(<i>x</i>, <i>t</i>) of (<span>\\(*\\)</span>) is globally defined and blows-up in infinite time. As <span>\\(t\\rightarrow +\\infty \\)</span> it has the approximate profile </p><div><div><span>$$\\begin{aligned} u(x,t) \\approx \\frac{1}{\\lambda ^2(t)} U\\left( \\frac{x-\\xi (t)}{\\lambda (t)} \\right) , \\quad U(y)= \\frac{8}{(1+|y|^2)^2}, \\end{aligned}$$</span></div></div><p>where <span>\\(\\lambda (t) \\approx \\frac{c}{\\sqrt{\\log t}}\\)</span>, <span>\\(\\xi (t)\\rightarrow q\\)</span> for some <span>\\(c>0\\)</span> and <span>\\(q\\in {\\mathbb {R}}^2\\)</span>. This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6000,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02006-7.pdf","citationCount":"0","resultStr":"{\"title\":\"Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System\",\"authors\":\"Juan Dávila, Manuel del Pino, Jean Dolbeault, Monica Musso, Juncheng Wei\",\"doi\":\"10.1007/s00205-024-02006-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system </p><div><figure><div><div><picture><img></picture></div></div></figure></div><p> We consider the critical mass case <span>\\\\(\\\\int _{{\\\\mathbb {R}}^2} u_0(x)\\\\, \\\\textrm{d}x = 8\\\\pi \\\\)</span>, which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function <span>\\\\(u_0^*\\\\)</span> with mass <span>\\\\(8\\\\pi \\\\)</span> such that for any initial condition <span>\\\\(u_0\\\\)</span> sufficiently close to <span>\\\\(u_0^*\\\\)</span> and mass <span>\\\\(8\\\\pi \\\\)</span>, the solution <i>u</i>(<i>x</i>, <i>t</i>) of (<span>\\\\(*\\\\)</span>) is globally defined and blows-up in infinite time. As <span>\\\\(t\\\\rightarrow +\\\\infty \\\\)</span> it has the approximate profile </p><div><div><span>$$\\\\begin{aligned} u(x,t) \\\\approx \\\\frac{1}{\\\\lambda ^2(t)} U\\\\left( \\\\frac{x-\\\\xi (t)}{\\\\lambda (t)} \\\\right) , \\\\quad U(y)= \\\\frac{8}{(1+|y|^2)^2}, \\\\end{aligned}$$</span></div></div><p>where <span>\\\\(\\\\lambda (t) \\\\approx \\\\frac{c}{\\\\sqrt{\\\\log t}}\\\\)</span>, <span>\\\\(\\\\xi (t)\\\\rightarrow q\\\\)</span> for some <span>\\\\(c>0\\\\)</span> and <span>\\\\(q\\\\in {\\\\mathbb {R}}^2\\\\)</span>. This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).</p></div>\",\"PeriodicalId\":55484,\"journal\":{\"name\":\"Archive for Rational Mechanics and Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2024-06-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00205-024-02006-7.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Archive for Rational Mechanics and Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00205-024-02006-7\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-024-02006-7","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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(x, t) of (\(*\)) is globally defined and blows-up in infinite time. As \(t\rightarrow +\infty \) it has the approximate profile
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).
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.