{"title":"Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space","authors":"K. M. Hui, Jinwan Park","doi":"10.3934/DCDS.2021085","DOIUrl":null,"url":null,"abstract":"We study the existence, uniqueness, and asymptotic behaviour of the singular solution of the fast diffusion equation, which blows up at the origin for all time. For $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $\\beta<0$ and $\\alpha=\\frac{2\\beta}{1-m}$, we prove the existence and asymptotic behaviour of singular eternal self-similar solution of the fast diffusion equation. As a consequence, we prove the existence and uniqueness of solution of Cauchy problem for the fast diffusion equation. \r\nFor $n=3, 4$ and $\\frac{n-2}{n+2}\\le m 0$. Furthermore, for the radially symmetric initial value $u_0$, $3 \\le n < 8$, $1- \\sqrt{\\frac{2}{n}} \\le m \\le \\min \\left \\{\\frac{2(n-2)}{3n}, \\frac{n-2}{n+2}\\right \\}$, we also have the asymptotic large time behaviour.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Analysis of PDEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/DCDS.2021085","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
We study the existence, uniqueness, and asymptotic behaviour of the singular solution of the fast diffusion equation, which blows up at the origin for all time. For $n\ge 3$, $0
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