{"title":"时变域上半线性热方程的连续性和回拉吸引子","authors":"Mingli Hong, Feng Zhou, Chunyou Sun","doi":"10.1186/s13661-023-01813-3","DOIUrl":null,"url":null,"abstract":"We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term $f(\\cdot )$ to satisfy $\\int _{-\\infty}^{t}e^{\\lambda s}\\|f(s)\\|^{2}_{L^{2}}\\,ds<\\infty $ for all $t\\in \\mathbb{R}$ , we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in $H^{1}$ topology and the usual $(L^{2},L^{2})$ pullback $\\mathscr{D}_{\\lambda}$ -attractor indeed can attract in the $H^{1}$ -norm, provided that $\\int _{-\\infty}^{t}e^{\\lambda s}\\|f(s)\\|^{2}_{H^{-1}(\\mathcal{O}_{s})}\\,ds< \\infty $ and $f\\in L^{2}_{\\mathrm{loc}}(\\mathbb{R},L^{2}(\\mathcal{O}_{s}))$ .","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Continuity and pullback attractors for a semilinear heat equation on time-varying domains\",\"authors\":\"Mingli Hong, Feng Zhou, Chunyou Sun\",\"doi\":\"10.1186/s13661-023-01813-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term $f(\\\\cdot )$ to satisfy $\\\\int _{-\\\\infty}^{t}e^{\\\\lambda s}\\\\|f(s)\\\\|^{2}_{L^{2}}\\\\,ds<\\\\infty $ for all $t\\\\in \\\\mathbb{R}$ , we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in $H^{1}$ topology and the usual $(L^{2},L^{2})$ pullback $\\\\mathscr{D}_{\\\\lambda}$ -attractor indeed can attract in the $H^{1}$ -norm, provided that $\\\\int _{-\\\\infty}^{t}e^{\\\\lambda s}\\\\|f(s)\\\\|^{2}_{H^{-1}(\\\\mathcal{O}_{s})}\\\\,ds< \\\\infty $ and $f\\\\in L^{2}_{\\\\mathrm{loc}}(\\\\mathbb{R},L^{2}(\\\\mathcal{O}_{s}))$ .\",\"PeriodicalId\":49228,\"journal\":{\"name\":\"Boundary Value Problems\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2024-01-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Boundary Value Problems\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1186/s13661-023-01813-3\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Boundary Value Problems","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1186/s13661-023-01813-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
Continuity and pullback attractors for a semilinear heat equation on time-varying domains
We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term $f(\cdot )$ to satisfy $\int _{-\infty}^{t}e^{\lambda s}\|f(s)\|^{2}_{L^{2}}\,ds<\infty $ for all $t\in \mathbb{R}$ , we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in $H^{1}$ topology and the usual $(L^{2},L^{2})$ pullback $\mathscr{D}_{\lambda}$ -attractor indeed can attract in the $H^{1}$ -norm, provided that $\int _{-\infty}^{t}e^{\lambda s}\|f(s)\|^{2}_{H^{-1}(\mathcal{O}_{s})}\,ds< \infty $ and $f\in L^{2}_{\mathrm{loc}}(\mathbb{R},L^{2}(\mathcal{O}_{s}))$ .
期刊介绍:
The main aim of Boundary Value Problems is to provide a forum to promote, encourage, and bring together various disciplines which use the theory, methods, and applications of boundary value problems. Boundary Value Problems will publish very high quality research articles on boundary value problems for ordinary, functional, difference, elliptic, parabolic, and hyperbolic differential equations. Articles on singular, free, and ill-posed boundary value problems, and other areas of abstract and concrete analysis are welcome. In addition to regular research articles, Boundary Value Problems will publish review articles.