( L 2 , L p ) -random attractors for stochastic reaction–diffusion equation on unbounded domains

• Published in: Nonlinear analysis Vol. 75; no. 2; pp. 485 - 502
• Main Authors: Zhao, Wenqiang, Li, Yangrong
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 2012; Elsevier
• Subjects: Asymptotic compactness; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Ordinary differential equations; Partial differential equations; Random attractor; Random dynamical systems; Sciences and techniques of general use; Stochastic reaction–diffusion equation; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Stochastic reaction–diffusion equation; Random attractor; Random dynamical systems; Asymptotic compactness; 35B41; 35B40; 60H15; Stochastic reaction-diffusion equation; Nonlinear analysis; Dynamical system; Reaction diffusion equation; Attractor; Mathematical model; Stochastic equation
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2011.08.050

In this paper, the existence of ( L 2 , L p ) -random attractor is established for a stochastic reaction–diffusion equation on the whole space R N . This random attractor is a compact and invariant tempered set which attracts every tempered random subset of L 2 in the topology of L p . The nonlinearity f is supposed to satisfy some growth of arbitrary order p − 1 , where p ≥ 2 . The ( L 2 , L p ) -asymptotic compactness of the random dynamical system is proved by an asymptotic a priori estimate of the unbounded part of solutions.