{"title":"具有非线性彩色噪声的随机准线性方程的回拉随机吸引子的高阶连续性","authors":"Yangrong Li, Fengling Wang, Tomás Caraballo","doi":"10.1007/s10884-023-10333-8","DOIUrl":null,"url":null,"abstract":"<p>For a nonautonomous random dynamical system, we introduce a concept of a pullback random bi-spatial attractor (PRBA). We prove an existence theorem of a PRBA, which includes its measurability, compactness and attraction in the regular space. We then establish the residual dense continuity of a family of PRBAs from a parameter space into the space of all compact subsets of the regular space equipped by Hausdorff metric. The abstract results are illustrated in the nonautonomous random quasilinear equation driven by nonlinear colored noise, where the size of noise belongs to <span>\\((0,\\infty ]\\)</span> and the infinite size corresponds to the deterministic equation. The application results are the existence and residual dense continuity of PRBAs on <span>\\((0,\\infty ]\\)</span> in both square and <i>p</i>-order Lebesgue spaces, where <span>\\(p>2\\)</span>. The lower semi-continuity of attractors in the regular space seems to be a new subject even for an autonomous deterministic system.</p>","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"82 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Higher-Order Continuity of Pullback Random Attractors for Random Quasilinear Equations with Nonlinear Colored Noise\",\"authors\":\"Yangrong Li, Fengling Wang, Tomás Caraballo\",\"doi\":\"10.1007/s10884-023-10333-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For a nonautonomous random dynamical system, we introduce a concept of a pullback random bi-spatial attractor (PRBA). We prove an existence theorem of a PRBA, which includes its measurability, compactness and attraction in the regular space. We then establish the residual dense continuity of a family of PRBAs from a parameter space into the space of all compact subsets of the regular space equipped by Hausdorff metric. The abstract results are illustrated in the nonautonomous random quasilinear equation driven by nonlinear colored noise, where the size of noise belongs to <span>\\\\((0,\\\\infty ]\\\\)</span> and the infinite size corresponds to the deterministic equation. The application results are the existence and residual dense continuity of PRBAs on <span>\\\\((0,\\\\infty ]\\\\)</span> in both square and <i>p</i>-order Lebesgue spaces, where <span>\\\\(p>2\\\\)</span>. The lower semi-continuity of attractors in the regular space seems to be a new subject even for an autonomous deterministic system.</p>\",\"PeriodicalId\":15624,\"journal\":{\"name\":\"Journal of Dynamics and Differential Equations\",\"volume\":\"82 1\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-01-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Dynamics and Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10884-023-10333-8\",\"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":"Journal of Dynamics and Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10884-023-10333-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Higher-Order Continuity of Pullback Random Attractors for Random Quasilinear Equations with Nonlinear Colored Noise
For a nonautonomous random dynamical system, we introduce a concept of a pullback random bi-spatial attractor (PRBA). We prove an existence theorem of a PRBA, which includes its measurability, compactness and attraction in the regular space. We then establish the residual dense continuity of a family of PRBAs from a parameter space into the space of all compact subsets of the regular space equipped by Hausdorff metric. The abstract results are illustrated in the nonautonomous random quasilinear equation driven by nonlinear colored noise, where the size of noise belongs to \((0,\infty ]\) and the infinite size corresponds to the deterministic equation. The application results are the existence and residual dense continuity of PRBAs on \((0,\infty ]\) in both square and p-order Lebesgue spaces, where \(p>2\). The lower semi-continuity of attractors in the regular space seems to be a new subject even for an autonomous deterministic system.
期刊介绍:
Journal of Dynamics and Differential Equations serves as an international forum for the publication of high-quality, peer-reviewed original papers in the field of mathematics, biology, engineering, physics, and other areas of science. The dynamical issues treated in the journal cover all the classical topics, including attractors, bifurcation theory, connection theory, dichotomies, stability theory and transversality, as well as topics in new and emerging areas of the field.