{"title":"无界域上分数随机延迟反应-扩散方程的长期动力学","authors":"Zhang Chen, Bixiang Wang","doi":"10.1007/s40072-024-00334-z","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we investigate the long-term dynamics of fractional stochastic delay reaction-diffusion equations on unbounded domains with a polynomial drift term of arbitrary order driven by nonlinear noise. We first define a mean random dynamical system in a Hilbert space for the solutions of the equation and prove the existence and uniqueness of weak pullback mean random attractors. We then establish the existence and regularity of invariant measures of the system under further conditions on the nonlinear delay and diffusion terms. We also prove the tightness of the set of all invariant measures of the equation when the time delay varies in a bounded interval. We finally show that every limit of a sequence of invariant measures of the delay equation must be an invariant measure of the limiting system as delay approaches zero. The uniform tail-estimates and the Ascoli–Arzelà theorem are used to derive the tightness of distribution laws of solutions in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.</p>","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Long-term dynamics of fractional stochastic delay reaction–diffusion equations on unbounded domains\",\"authors\":\"Zhang Chen, Bixiang Wang\",\"doi\":\"10.1007/s40072-024-00334-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we investigate the long-term dynamics of fractional stochastic delay reaction-diffusion equations on unbounded domains with a polynomial drift term of arbitrary order driven by nonlinear noise. We first define a mean random dynamical system in a Hilbert space for the solutions of the equation and prove the existence and uniqueness of weak pullback mean random attractors. We then establish the existence and regularity of invariant measures of the system under further conditions on the nonlinear delay and diffusion terms. We also prove the tightness of the set of all invariant measures of the equation when the time delay varies in a bounded interval. We finally show that every limit of a sequence of invariant measures of the delay equation must be an invariant measure of the limiting system as delay approaches zero. The uniform tail-estimates and the Ascoli–Arzelà theorem are used to derive the tightness of distribution laws of solutions in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.</p>\",\"PeriodicalId\":48569,\"journal\":{\"name\":\"Stochastics and Partial Differential Equations-Analysis and Computations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-06-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Stochastics and Partial Differential Equations-Analysis and Computations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s40072-024-00334-z\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastics and Partial Differential Equations-Analysis and Computations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40072-024-00334-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Long-term dynamics of fractional stochastic delay reaction–diffusion equations on unbounded domains
In this paper, we investigate the long-term dynamics of fractional stochastic delay reaction-diffusion equations on unbounded domains with a polynomial drift term of arbitrary order driven by nonlinear noise. We first define a mean random dynamical system in a Hilbert space for the solutions of the equation and prove the existence and uniqueness of weak pullback mean random attractors. We then establish the existence and regularity of invariant measures of the system under further conditions on the nonlinear delay and diffusion terms. We also prove the tightness of the set of all invariant measures of the equation when the time delay varies in a bounded interval. We finally show that every limit of a sequence of invariant measures of the delay equation must be an invariant measure of the limiting system as delay approaches zero. The uniform tail-estimates and the Ascoli–Arzelà theorem are used to derive the tightness of distribution laws of solutions in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.
期刊介绍:
Stochastics and Partial Differential Equations: Analysis and Computations publishes the highest quality articles presenting significantly new and important developments in the SPDE theory and applications. SPDE is an active interdisciplinary area at the crossroads of stochastic anaylsis, partial differential equations and scientific computing. Statistical physics, fluid dynamics, financial modeling, nonlinear filtering, super-processes, continuum physics and, recently, uncertainty quantification are important contributors to and major users of the theory and practice of SPDEs. The journal is promoting synergetic activities between the SPDE theory, applications, and related large scale computations. The journal also welcomes high quality articles in fields strongly connected to SPDE such as stochastic differential equations in infinite-dimensional state spaces or probabilistic approaches to solving deterministic PDEs.