{"title":"刺穿欧几里得空间中快速扩散方程奇异解的渐近行为","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":"27 3 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"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\":\"27 3 1\",\"pages\":\"\"},\"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}","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}
Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space
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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