{"title":"半线性抛物方程的一个平凡结果","authors":"G. Catino, D. Castorina, C. Mantegazza","doi":"10.3934/MINE.2022002","DOIUrl":null,"url":null,"abstract":"We show a triviality result for \"pointwise\" monotone in time, bounded \"eternal\" solutions of the semilinear heat equation \\begin{equation*} u_{t}=\\Delta u + |u|^{p} \\end{equation*} on complete Riemannian manifolds of dimension $n \\geq 5$ with nonnegative Ricci tensor, when $p$ is smaller than the critical Sobolev exponent $\\frac{n+2}{n-2}$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"45 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"A triviality result for semilinear parabolic equations\",\"authors\":\"G. Catino, D. Castorina, C. Mantegazza\",\"doi\":\"10.3934/MINE.2022002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show a triviality result for \\\"pointwise\\\" monotone in time, bounded \\\"eternal\\\" solutions of the semilinear heat equation \\\\begin{equation*} u_{t}=\\\\Delta u + |u|^{p} \\\\end{equation*} on complete Riemannian manifolds of dimension $n \\\\geq 5$ with nonnegative Ricci tensor, when $p$ is smaller than the critical Sobolev exponent $\\\\frac{n+2}{n-2}$.\",\"PeriodicalId\":8445,\"journal\":{\"name\":\"arXiv: Analysis of PDEs\",\"volume\":\"45 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-07-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Analysis of PDEs\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3934/MINE.2022002\",\"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/MINE.2022002","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 3
摘要
当$p$小于临界Sobolev指数$\frac{n+2}{n-2}$时，我们给出了半线性热方程\begin{equation*} u_{t}=\Delta u + |u|^{p} \end{equation*}在具有非负Ricci张量的$n \geq 5$维完全黎曼流形上的“点向”单调时间上的有界“永恒”解的平凡性结果。
A triviality result for semilinear parabolic equations
We show a triviality result for "pointwise" monotone in time, bounded "eternal" solutions of the semilinear heat equation \begin{equation*} u_{t}=\Delta u + |u|^{p} \end{equation*} on complete Riemannian manifolds of dimension $n \geq 5$ with nonnegative Ricci tensor, when $p$ is smaller than the critical Sobolev exponent $\frac{n+2}{n-2}$.
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