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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]

This paper deals with the asymptotic behavior of the non-autonomous random dynamical systems generated by the wave equations with supercritical nonlinearity driven by colored noise defined on R n R n with n ≤ 6 n ≤ 6. Based on the uniform Strichartz estimates, we prove the well-posedness of the equation in the natural energy space and define a continuous cocycle associated with the solution operator. We also establish the existence and uniqueness of tempered random attractors of the equation by showing the uniform smallness of the tails of the solutions outside a bounded domain in order to overcome the non-compactness of Sobolev embeddings on unbounded domains. [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 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]

The random dynamics in Hs(Rn) with s ∈ (0,1) is investigated for the fractional nonclassical diffusion equations driven by colored noise. Both existence and uniqueness of pullback random attractors are established for the equations with a wide class of nonlinear diffusion terms. In the case of additive noise, the upper semi-continuity of these attractors is proved as the correlation time of the colored noise approaches zero. The methods of uniform tail-estimate and spectral decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embedding on an unbounded domain. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic behavior of the solutions of the non-autonomous one-dimensional stochastic Keller-Segel equations defined in a bounded interval with Neumann boundary conditions. We prove the existence and uniqueness of tempered pullback random attractors under certain conditions. We also establish the convergence of the solutions as well as the pullback random attractors of the stochastic equations as the intensity of noise approaches zero. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of a class of non-autonomous non-local fractional stochastic parabolic equation driven by multiplicative white noise on the entire space ℝn. We first prove the pathwise well-posedness of the equation and define a continuous non-autonomous cocycle in L²(ℝn). We then prove the existence and uniqueness of tempered pullback attractors for the cocycle under certain dissipative conditions. The periodicity of the tempered attractors is also proved when the deterministic non-autonomous external terms are periodic in time. The pullback asymptotic compactness of the cocycle in L²(ℝn) is established by the uniform estimates on the tails of solutions for sufficiently large space and time variables. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic behavior of the solutions of the two-dimensional stochastic Navier-Stokes equations via the stationary Wong-Zakai approximations given by the Wiener shift. We prove the existence and uniqueness of tempered pullback attractors for the random equations of the Wong-Zakai approximations with a Lipschitz continuous diffusion term. Under certain conditions, we also prove the convergence of solutions and random attractors of the approximate equations when the step size of approximations approaches zero. [ABSTRACT FROM AUTHOR]

We first apply the Galerkin method to prove the existence and uniqueness of solutions for a class of non-autonomous fractional reaction–diffusion equations driven by multiplicative noise. We then establish the existence and uniqueness of tempered pullback random attractors for the equations in an appropriate Hilbert space. The upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero. [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]

Abstract: In this paper, we study the existence, stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise. We first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions. We then examine the existence and stability of random complete quasi-solutions and establish the relations of these solutions and the structures of tempered attractors. When the stochastic equations are incorporated with periodic forcing, we obtain the existence and stability of random periodic solutions. For the stochastic Chafee–Infante equation, we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions. [Copyright &y& Elsevier]

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: 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: 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 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]

In this paper, we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of the stochastic reaction–diffusion equation driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong–Zakai approximations of stochastic reaction–diffusion equation. Then, we show that the attractors of Wong–Zakai approximations converges to the attractor of the stochastic reaction–diffusion equation for both additive and multiplicative noise. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions to a class of non-autonomous stochastic lattice systems driven by multiplicative white noise. We prove the existence and uniqueness of tempered random attractors in a weighted space containing all bounded sequences, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero. We also construct maximal and minimal tempered random complete solutions which bound the attractors from above and below, respectively. When deterministic external forcing terms are periodic in time, we show the random attractors are pathwise periodic. In addition, we exhibit a non-autonomous stochastic lattice system which possesses an infinite-dimensional tempered random attractor. [ABSTRACT FROM AUTHOR]

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]

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]