{"title":"Dynamics and Wong-Zakai Approximations of Stochastic Nonlocal PDEs with Long Time Memory","authors":"Jiaohui Xu, Tomás Caraballo, José Valero","doi":"10.1007/s12346-024-01080-2","DOIUrl":null,"url":null,"abstract":"<p>In this paper, a combination of Galerkin’s method and Dafermos’ transformation is first used to prove the existence and uniqueness of solutions for a class of stochastic nonlocal PDEs with long time memory driven by additive noise. Next, the existence of tempered random attractors for such equations is established in an appropriate space for the analysis of problems with delay and memory. Eventually, the convergence of solutions of Wong-Zakai approximations and upper semicontinuity of random attractors of the approximate random system, as the step sizes of approximations approach zero, are analyzed in a detailed way.</p>","PeriodicalId":48886,"journal":{"name":"Qualitative Theory of Dynamical Systems","volume":"30 1","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Qualitative Theory of Dynamical Systems","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s12346-024-01080-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, a combination of Galerkin’s method and Dafermos’ transformation is first used to prove the existence and uniqueness of solutions for a class of stochastic nonlocal PDEs with long time memory driven by additive noise. Next, the existence of tempered random attractors for such equations is established in an appropriate space for the analysis of problems with delay and memory. Eventually, the convergence of solutions of Wong-Zakai approximations and upper semicontinuity of random attractors of the approximate random system, as the step sizes of approximations approach zero, are analyzed in a detailed way.
期刊介绍：
Qualitative Theory of Dynamical Systems (QTDS) publishes high-quality peer-reviewed research articles on the theory and applications of discrete and continuous dynamical systems. The journal addresses mathematicians as well as engineers, physicists, and other scientists who use dynamical systems as valuable research tools. The journal is not interested in numerical results, except if these illustrate theoretical results previously proved.
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