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A system of stochastic retarded reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing on thin domains is considered. Relations between the asymptotic behavior for the stochastic retarded equations defined on thin domains in Rn+1 and an equation on a domain in Rn are investigated. We first show the existence and uniqueness of tempered random attractors for these equations. Then, we analyze convergence properties of the solutions as well as the attractors. [ABSTRACT FROM AUTHOR]

This paper deals with the limiting behavior of stochastic reaction–diffusion equations driven by multiplicative noise and deterministic non-autonomous terms defined on thin domains. We first prove the existence, uniqueness and periodicity of pullback tempered random attractors for the equations in an ( n + 1 ) -dimensional narrow domain, and then establish the upper semicontinuity of these attractors when a family of ( n + 1 ) -dimensional thin domains collapses onto an n -dimensional domain. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of tempered random attractors for stochastic Reaction-Diffusion equations on unbounded domains with multiplicative noise and deterministic non-autonomous forcing. We establish the periodicity of the tempered attractors when the stochastic equations are forced by periodic functions. We further prove the upper semicontinuity of these attractors when the intensity of stochastic perturbations approaches zero. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with pullback attractors of the stochastic p-Laplace equation defined on the entire space . We first establish the asymptotic compactness of the equation in and then prove the existence and uniqueness of non-autonomous random attractors. This attractor is pathwise periodic if the non-autonomous deterministic forcing is time periodic. The difficulty of non-compactness of Sobolev embeddings on is overcome by the uniform smallness of solutions outside a bounded domain. [Copyright &y& Elsevier]

Abstract: We study pullback attractors of non-autonomous non-compact dynamical systems generated by differential equations with non-autonomous deterministic as well as stochastic forcing terms. We first introduce the concepts of pullback attractors and asymptotic compactness for such systems. We then prove a sufficient and necessary condition for existence of pullback attractors. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors. For random systems containing periodic deterministic forcing terms, we show the pullback attractors are also periodic under certain conditions. As an application of the abstract theory, we prove the existence of a unique pullback attractor for Reaction–Diffusion equations on with both deterministic and random external terms. Since Sobolev embeddings are not compact on unbounded domains, the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result to the Chafee–Infante equation is discussed. [Copyright &y& Elsevier]

Abstract: We study the asymptotic behavior of solutions of a class of singularly perturbed non-autonomous parabolic equations defined on with unbounded external terms. We first prove the existence of a pullback attractor for the perturbed equation in and then establish the upper semicontinuity of these attractors as small perturbations approach zero. The uniform estimates on the tails of solutions are used to overcome the difficulty caused by the non-compactness of Sobolev embeddings on unbounded domains. [Copyright &y& Elsevier]

Abstract: We investigate the asymptotic behavior of solutions of a class of non-autonomous degenerate parabolic equations on with unbounded external forcing terms. The existence of a pullback global attractor is proved in for any . [Copyright &y& Elsevier]

Abstract: The existence of a random attractor for the stochastic FitzHugh–Nagumo system defined on an unbounded domain is established. The pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique. [Copyright &y& Elsevier]

We study the long time behavior of solutions of the non-autonomous reaction-diffusion equation defined on the entire space ℝ n when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L 2(ℝ n ) and H 1(ℝ n ), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains. [ABSTRACT FROM AUTHOR]

We study the long time behavior of solutions of the non-autonomous reaction-diffusion equation defined on the entire space ℝ n when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L 2(ℝ n ) and H 1(ℝ n ), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains. [ABSTRACT FROM AUTHOR]

Abstract: The existence of a pullback attractor is established for the singularly perturbed FitzHugh–Nagumo system defined on the entire space when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero. [Copyright &y& Elsevier]

Abstract: We prove the existence of a compact random attractor for the stochastic Benjamin–Bona–Mahony equation defined on an unbounded domain. This random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity. [Copyright &y& Elsevier]

We investigate the asymptotic behavior of solutions of the p -Laplace equation driven simultaneously by non-autonomous deterministic forcing and multiplicative white noise on R n . We show the tails of solutions of the equation are uniformly small outside a bounded domain, which is used to derive asymptotic compactness of solution operators in L 2 ( R n ) by overcoming the non-compactness of Sobolev embeddings on unbounded domains. We then prove existence and uniqueness of random attractors and further establish upper semicontinuity of attractors as the intensity of noise approaches zero. [ABSTRACT FROM AUTHOR]

Abstract: The existence of a pullback attractor is established for a stochastic reaction–diffusion equation on all n-dimensional space. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise. The reaction–diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far-field values of solutions. [Copyright &y& Elsevier]