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In this paper, we study the asymptotic behavior of solutions of fractional nonclassical diffusion equations with delay driven by additive noise defined on unbounded domains. We first prove the uniform compactness of pullback random attractors of the equation with respect to noise intensity and time delay, and then establish the upper semi-continuity of these attractors as either noise intensity or time delay approaches zero. [ABSTRACT FROM AUTHOR]

The global well-posedness as well as long-term behavior in terms of mean random attractors and invariant measures are investigated for a class of stochastic discrete reaction-diffusion equations defined on Z k with a family of superlinear noise. The existence and uniqueness of weak pullback mean random attractors for the mean random dynamical system associated with the non-autonomous equations are established in L 2 (Ω , ℓ 2). The existence of invariant measures for the autonomous equations is established in ℓ 2 by Krylov-Bogolyubov's method. The idea of uniform estimates on the tails of solutions is employed to establish the tightness of a family of distribution laws of the solutions. It seems that this is the first time to study the random attractors and invariant measures of stochastic equations with superlinear noise. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of long term behavior of the two-dimensional random Navier-Stokes equations driven by colored noise defined in bounded and unbounded domains. We prove the existence and uniqueness of pullback random attractors for the equations with Lipschitz diffusion terms. In the case of additive noise, we show the upper semi-continuity of these attractors when the correlation time of the colored noise approaches zero. When the equations are defined on unbounded domains, we establish the pullback asymptotic compactness of the solutions by Ball's idea of energy equations in order to overcome the difficulty introduced by the noncompactness of Sobolev embeddings. [ABSTRACT FROM AUTHOR]

We study the random dynamics of the N-dimensional stochastic Schrödinger lattice systems with locally Lipschitz diffusion terms driven by locally Lipschitz nonlinear noise. We first prove the existence and uniqueness of solutions and define a mean random dynamical system associated with the solution operators. We then establish the existence and uniqueness of weak pullback random attractors in a Bochner space. We finally prove the existence of invariant measures of the stochastic equation in the space of complex-valued square-summable sequences. The tightness of a family of probability distributions of solutions is derived by the uniform estimates on the tails of the solutions at far field. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the solutions of the fractional reaction-diffusion equations with polynomial drift terms of arbitrary order driven by locally Lipschitz nonlinear diffusion terms defined on R n. We first prove the well-posedness of the equation based on pathwise uniform estimates as well as uniform estimates on average. We then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors. We finally establish the existence of invariant measures when the diffusion terms are globally Lipschitz continuous. The uniform estimates on the tails of solutions are employed to prove the tightness of a family of probability distributions of solutions in order to overcome the non-compactness of usual Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]

This paper is concerned with weak pullback mean random attractors for mean random dynamical systems defined in Bochner spaces. We first introduce the concept of weak pullback mean random attractor with respect to the weak topology of reflexive Bochner spaces and then provide a sufficient criterion for existence and uniqueness of such attractors over a complete filtered probability space. As an application, we prove the existence and uniqueness of weak pullback mean random attractors for the stochastic reaction–diffusion equations with nonlinear drift terms as well as nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic reaction–diffusion equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time. [ABSTRACT FROM AUTHOR]

This article is concerned with the asymptotic behavior of solutions of a wide class of non-autonomous, non-local, fractional, stochastic p -Laplacian equations driven by multiplicative white noise. The time-dependent nonlinear drift term of the equation has a polynomial growth of arbitrary order in its third component which is allowed to be greater than the exponent p. We first employ the Faedo–Galerkin method to prove the well-posedness of the equation in an appropriate Hilbert space. We then establish the existence, uniqueness and periodicity of tempered pullback random attractors for the equations. The upper semi-continuity of these attractors is also derived as the density of noise tends to zero. The results of this paper are new even when the stochastic equation reduces to the deterministic one. [ABSTRACT FROM AUTHOR]

We investigate the pathwise asymptotic behavior of the FitzHugh–Nagumo systems defined on unbounded domains driven by nonlinear colored noise. We prove the existence and uniqueness of tempered pullback random attractors of the systems with polynomial diffusion terms. The pullback asymptotic compactness of solutions is obtained by the uniform estimates on the tails of solutions outside a bounded domain. We also examine the limiting behavior of the FitzHugh–Nagumo systems driven by linear colored noise as the correlation time of the colored noise approaches zero. In this respect, we prove that the solutions and the pullback random attractors of the systems driven by linear colored noise converge to that of the corresponding stochastic systems driven by linear white noise. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the solutions of the nonautonomous fractional stochastic reaction-diffusion equations on R n with continuous (but not necessarily differentiable) nonlinear drift terms, which leads to the nonuniqueness of solutions and hence a multivalued random dynamical system with the solution operators of the equation. We first show the existence and uniqueness of random attractors for such a dynamical system and then establish the upper semicontinuity of these attractors as the intensity of noise approaches zero. The measurability of the random attractors is proved by the method of weak upper semicontinuity of multivalued functions, and the pullback asymptotic compactness of the system is derived by the idea of uniform estimates on the tails of the solutions. [ABSTRACT FROM AUTHOR]

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 is concerned with the well-posedness and long term behavior of the non-autonomous random wave equations driven by nonlinear colored noise on R n with n ≤ 5. The drift nonlinearity has a supercritical growth exponent (n + 2) / (n − 2). We first prove the existence and uniqueness of solutions in the energy space by showing the non-concentration of energy via the Morawetz identity and the uniform Strichartz estimates. We then prove the existence and uniqueness of tempered pullback random attractors of the non-autonomous random dynamical system associated with the equation. The asymptotic compactness of solutions is obtained by the idea of energy equation due to Ball and the uniform tail-ends estimates in order to circumvent the difficulty caused by the lack of compactness of Sobolev embeddings on unbounded domains. [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]

This paper deals with the asymptotic behavior of solutions of parabolic equations on unbounded domains which contain stochastic terms as well as non-autonomous deterministic terms. We define a continuous random dynamical system for the equations in weighted Sobolev spaces that allow functions to have certain polynomial growth rate at infinity and hence include all bounded solutions. We prove pullback asymptotic compactness of solutions as well as the existence and uniqueness of tempered random attractors in the weighted spaces. The structures of the tempered attractors are fully characterized by tempered complete solutions. In the case where the non-autonomous deterministic terms are periodic in time, we show that the tempered random attractors are also periodic in time. To overcome the difficulty of non-compactness of Sobolev embeddings on unbounded domains, the idea of uniform pathwise estimates on the tails of solutions is employed to show asymptotic compactness. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the asymptotic behavior of solutions of a stochastic nonlinear wave equation with dispersive and dissipative terms defined on an unbounded domain. It is proved that the random dynamical system generated by the equation has a random attractor in a Sobolev space. To overcome the difficulty caused by the non-compactness of Sobolev embeddings on unbounded domains, a cut-off method and a decomposition trick are combined to prove the asymptotic compactness of the solutions. [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: 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]

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]

Abstract: We prove the existence of a global attractor for the Newton–Boussinesq equation defined in a two-dimensional channel. The asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions. We also establish the regularity of the global attractor. [Copyright &y& Elsevier]

Abstract: The FitzHugh–Nagumo system on infinite lattices is studied. By means of “tail ends” estimates on solutions, it is proved that the system is asymptotically compact in a weighted space and has a compact global attractor containing traveling wave solutions. The singular limiting behavior of global attractors is also investigated as a singular parameter . It is shown that the limiting system for has no global attractor, but all the global attractors for perturbed systems are contained in a common compact subset of the phase space when is positive but small. Further, a compact local attractor for the limiting system is constructed, and the upper semicontinuity of global attractors is established when . [Copyright &y& Elsevier]

We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in H s ( R n ) with s ∈ ( 0 , 1 ) . We prove the existence and uniqueness of the tempered random attractor that is compact in H s ( R n ) and attracts all tempered random subsets of L 2 ( R n ) with respect to the norm of H s ( R n ) . The main difficulty is to show the pullback asymptotic compactness of solutions in H s ( R n ) due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains. [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]

In this paper, we study the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for the stochastic lattice systems. We first prove that the approximate system has a unique tempered pullback attractor under much weaker conditions than the original system. When the stochastic system is driven by a linear multiplicative noise or additive white noise, we prove the convergence of solutions and the upper semicontinuity of random attractors for Wong-Zakai approximations as the step-length of the Wiener shift approaches zero. [ABSTRACT FROM AUTHOR]

This article is concerned with the asymptotic behavior of solutions of a class of non-autonomous stochastic Fitzhugh-Nagumo systems driven by multiplicative white noise on unbounded thin domains. The existence and uniqueness of tempered pullback random attractors are proved for the systems defined on (n + 1) − dimensional unbounded narrow domains. We also establish the upper semicontinuity of these attractors when a family of (n + 1) − dimensional unbounded thin domains collapses onto an n −dimensional unbounded domain. A cut-off method and a decomposition technique are employed to derive the pullback asymptotic compactness of solutions in order to overcome the difficulties caused by the non-compactness of Sobolev embeddings on unbounded domains as well as the lack of regularity of one component of the solutions. [ABSTRACT FROM AUTHOR]