{"title":"具有整体空间记忆的退化抛物方程的动力学行为","authors":"Rong Guo, Xuan Leng","doi":"10.1186/s13661-024-01824-8","DOIUrl":null,"url":null,"abstract":"This paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on $\\mathbb{R}^{n}$ . Since the corresponding equation includes the degenerate term $\\operatorname{div}\\{a(x)\\nabla u\\}$ , it requires us to give appropriate assumptions about the weight function $a(x)$ for studying our problem. Based on this, we first obtain the existence of a bounded absorbing set, then verify the asymptotic compactness of a solution semigroup via the asymptotic contractive semigroup method. Finally, the existence and uniqueness of global attractors are proved. In particular, the nonlinearity f satisfies the polynomial growth of arbitrary order $p-1$ ( $p\\geq 2$ ) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Dynamical behavior of a degenerate parabolic equation with memory on the whole space\",\"authors\":\"Rong Guo, Xuan Leng\",\"doi\":\"10.1186/s13661-024-01824-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on $\\\\mathbb{R}^{n}$ . Since the corresponding equation includes the degenerate term $\\\\operatorname{div}\\\\{a(x)\\\\nabla u\\\\}$ , it requires us to give appropriate assumptions about the weight function $a(x)$ for studying our problem. Based on this, we first obtain the existence of a bounded absorbing set, then verify the asymptotic compactness of a solution semigroup via the asymptotic contractive semigroup method. Finally, the existence and uniqueness of global attractors are proved. In particular, the nonlinearity f satisfies the polynomial growth of arbitrary order $p-1$ ( $p\\\\geq 2$ ) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions.\",\"PeriodicalId\":49228,\"journal\":{\"name\":\"Boundary Value Problems\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2024-01-17\",\"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/s13661-024-01824-8\",\"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/s13661-024-01824-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
Dynamical behavior of a degenerate parabolic equation with memory on the whole space
This paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on $\mathbb{R}^{n}$ . Since the corresponding equation includes the degenerate term $\operatorname{div}\{a(x)\nabla u\}$ , it requires us to give appropriate assumptions about the weight function $a(x)$ for studying our problem. Based on this, we first obtain the existence of a bounded absorbing set, then verify the asymptotic compactness of a solution semigroup via the asymptotic contractive semigroup method. Finally, the existence and uniqueness of global attractors are proved. In particular, the nonlinearity f satisfies the polynomial growth of arbitrary order $p-1$ ( $p\geq 2$ ) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions.
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.