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This paper is concerned with pullback measure attractors of the non-autonomous stochastic reaction-diffusion equations defined in thin domains. We first prove the existence and uniqueness of pullback measure attractors for the inhomogeneous Markov process associated with the stochastic equations. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback measure attractors. Then we establish the upper semi-continuity of these attractors as a family of thin domains collapses into a lower-dimensional domain. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large deviation principle of the non-local fractional stochastic reaction-diffusion equation with a polynomial drift of arbitrary degree driven by multiplicative noise defined on unbounded domains. We first prove the strong convergence of the solutions of a control equation with respect to the weak topology of controls, and then show the convergence in distribution of the solutions of the stochastic equation when the noise intensity approaches zero. We finally establish the large deviations of the stochastic equation by the weak convergence method. The main difficulty of the paper is caused by the non-compactness of Sobolev embeddings on unbounded domains, and the idea of uniform tail-ends estimates is employed to circumvent the obstacle in order to obtain the tightness of distribution laws of the stochastic equation and the precompactness of the control equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the long term dynamics of the stochastic Schrödinger delay lattice systems when the nonlinear drift and diffusion terms are both locally Lipschitz continuous. Based on the well-posedness of the system, we first prove the existence and uniqueness of weak pullback mean random attractors in a product Hilbert space. We then show the tightness of distribution laws of solutions and the existence of invariant measures. We further prove the set of all invariant measures of the delay system is tight and every limit point of invariant measures of the delay system must be an invariant measure of the limiting system as time delay approaches zero. The idea of uniform tail-estimates is employed to to establish the tightness of distributions of solutions as well as the set of invariant measures of the delay system. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic compactness of the solutions of the two-dimensional Navier-Stokes equations defined in unbounded channel-like domains. In order to overcome the non-compactness of Sobolev embeddings in unbounded domains, we establish the uniform tail-ends estimates of solutions by showing all the solutions are uniformly small outside a sufficiently large bounded domain for all large time and bounded initial data, which implies the asymptotic compactness and hence the existence of global attractors in the natural energy space. The uniform tail-ends estimates are derived by the scalar stream function of the Navier-Stokes equations which is distinct from the reaction-diffusion equation due to the divergence free condition. [ABSTRACT FROM AUTHOR]

This paper deals with the limiting behavior of random attractors of stochastic wave equations with supercritical drift driven by linear multiplicative white noise defined on unbounded domains. We first establish the uniform Strichartz estimates of the solutions with respect to noise intensity, and then prove the convergence of the solutions of the stochastic equations with respect to initial data as well as noise intensity. To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the uniform tail-ends estimates to truncate the solutions in a bounded domain and then employ a spectral decomposition to establish the pre-compactness of the collection of all random attractors. We finally prove the upper semicontinuity of random attractor as noise intensity approaches zero. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a uniform contraction principle for verifying the uniform large deviation principles of locally Hölder continuous maps in Banach spaces. We then show the local Hölder continuity of the solutions of a class of fractional parabolic equations with polynomial drift of any order defined on ℝ n . We finally establish the large deviation principle of the fractional stochastic equations uniformly with respect to bounded initial data, despite the solution operators are not compact due to the non-compactness of Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]

Abstract The quality of dehydrated vegetables is affected by the degree to which they are returned to their original state during rehydration (restorability). At present, whether this mechanism occurs at the cell-wall or cell-membrane level is unclear. This paper reviews the important factors affecting the mechanism of dehydration–rehydration, focusing on the analysis of the composition and structure of the cell wall and cell membrane, and summarizes the related detection and analytical techniques that can be used to explore the mechanisms of dehydration–rehydration at the cell-wall and cell-membrane levels. The integrity and permeability of the cell membrane affect water transport during the dehydration–rehydration process. The cell wall and cell membrane are supporting materials for tissue morphology. The arabinan side chains of the primary structure and fibers are important for water retention. Water transport may be classified as symplastic and apoplastic. Cell membrane disruption occurs with symbiotic transport but increases the drying rate. An in-depth analysis of the dehydration–rehydration mechanism of vegetables will help develop and improve their processing methods and inspire new applications. [ABSTRACT FROM AUTHOR]

This paper is concerned with periodic measures of fractional stochastic reaction-diffusion equations with variable time delay defined on unbounded domains. We first prove the existence of periodic measures of the Markov process associated with the equation by showing the weak compactness of distribution laws of a family of solutions based on the uniform estimates and the equicontinuity of solutions in probability. We then establish the regularity of periodic measures and prove the tightness of the set of all periodic measures of the equation for small noise intensity. As a result, any sequence of periodic measures possesses at least one limit point, and we further prove every such limit must be a periodic measure of the corresponding limiting system. Finally, under further assumptions on the nonlinear terms, we establish the uniqueness and the exponential mixing property of periodic measures in the sense of Wasserstein metric. All the results of the paper are also valid for invariant measures of the corresponding autonomous version of the stochastic equation with constant time delay. The idea of uniform tail estimates of solutions in probability is employed to overcome the difficulty introduced by the non-compactness of Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]

This paper deals with the limiting behavior of invariant measures of the stochastic delay lattice systems. Under certain conditions, we first show the existence of invariant measures of the systems and then establish the stability in distribution of the solutions. We finally prove that any limit point of a tight sequence of invariant measures of the stochastic delay lattice systems must be an invariant measure of the corresponding limiting system as the intensity of noise converges or the time-delay approaches zero. In particular, when the stochastic delay lattice systems are stable in distribution, we show the invariant measures of the perturbed systems converge to that of the limiting system. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence of invariant measures and their limiting behavior of the stochastic Schrödinger lattice systems with respect to noise intensity. We prove the set of all invariant measures of the stochastic systems is weakly compact when the noise intensity varies in a bounded interval. We further show any limit of a sequence of invariant measures of the perturbed systems must be an invariant measure of the limiting system. [ABSTRACT FROM AUTHOR]

This paper deals with invariant measures of fractional stochastic reaction–diffusion equations on unbounded domains with locally Lipschitz continuous drift and diffusion terms. We first prove the existence and regularity of invariant measures, and then show the tightness of the set of all invariant measures of the equation when the noise intensity varies in a bounded interval. We also prove that every limit of invariant measures of the perturbed systems is an invariant measure of the corresponding limiting system. Under further conditions, we establish the ergodicity and the exponentially mixing property of invariant measures. [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]

In this paper, existence of invariant measure is mainly investigated for a fractional stochastic delay reaction–diffusion equation defined on unbounded domains. We first establish the mean-square uniform smallness of the tails of the solutions in order to overcome the non-compactness of standard Sobolev embeddings on unbounded domains. We then show the weak compactness of a family of probability distributions of the solutions by combining the Ascoli–Arzelŕ theorem, the uniform tail-estimates as well as the technique of dyadic division. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence of invariant measures in the Wasserstein sense for fractional stochastic reaction–diffusion equations defined on unbounded domains as the noise intensity approaches zero. Based on uniform estimates of solutions, we prove the family of invariant measures of the stochastic equations converges to the invariant measure of the corresponding deterministic equations in terms of the Wasserstein metric. We also provide the rate of such convergence. [ABSTRACT FROM AUTHOR]

The periodic measures of the stochastic delay reaction-diffusion lattice systems are investigated. Under a general condition, we prove the existence of periodic measures when the time-dependent terms of the system are periodic in time. Under further assumptions on the nonlinear terms, we show the set of all periodic measures of the perturbed system is weakly compact. Finally, we prove every limit point of a sequence of periodic measures of the stochastic delay system must be a periodic measure of the limiting system as the noise intensity or the time delay goes to zero. [ABSTRACT FROM AUTHOR]

This article is concerned with the global existence and random dynamics of the non-autonomous p -Laplacian lattice system defined on the entire integer set driven by infinite-dimensional nonlinear noise. The existence and uniqueness of mean square solutions to the equations are proved when the nonlinear drift and diffusion terms are locally Lipschitz continuous. It is shown that the mean random dynamical system generated by the solution operators has a unique tempered weak pullback random attractor in a Bochner space. The existence of invariant measures for the stochastic equations in the space of square summable sequences is also established. The idea of uniform tail-estimates of solutions is employed to show the tightness of a family of distribution laws of the solutions. [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]

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 partial differential equations driven by a white noise. We prove that the approximate equation has a pullback random attractor under much weaker conditions than the original stochastic equation. When the stochastic partial differential equation is driven by a linear multiplicative noise or additive white noise, we prove the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dynamics of the stochastic reaction–diffusion lattice systems defined on the entire integer set driven by locally Lipschitz nonlinear noise. We prove the existence and uniqueness of solutions as well as weak pullback random attractors for the mean random dynamical systems generated by the solution operators. The existence of invariant measures in ℓ 2 for the stochastic equations is also established. The idea of uniform estimates on the tails of the solutions is employed to show the tightness of a family of probability distributions of the solutions. [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]

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 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]

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]

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: 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: 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]

We study global asymptotic behavior of Poisson–Nernst–Planck (PNP) systems for flow of two ion species through a narrow tubular-like membrane channel. As the radius of the cross-section of the three-dimensional tubular-like membrane channel approaches zero, a one-dimensional limiting PNP system is derived. This one-dimensional limiting system differs from previously studied one-dimensional PNP systems in that it encodes the defining geometry of the three-dimensional membrane channel. To justify this limiting process, we show that the global attractors of the three-dimensional PNP systems are upper semi-continuous as the radius of the channel tends to zero. [ABSTRACT FROM AUTHOR]

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]

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: This paper is concerned with the continuity of strong solutions of the Reaction–Diffusion equation with respect to initial conditions. It is proved that the solutions of the equation are continuous in with respect to initial data provided the nonlinearity satisfies a dissipative condition with polynomial growth order . [Copyright &y& Elsevier]

Abstract: The asymptotic behavior of non-autonomous infinite-dimensional lattice systems is studied. It is shown that the non-autonomous lattice reaction–diffusion system has a compact uniform attractor. The uniform asymptotic compactness of the system is established by showing that the tails of the solutions are uniformly small when time goes to infinity. The upper semicontinuity of uniform attractors is also obtained when the infinite-dimensional reaction–diffusion system is approached by a family of finite-dimensional systems. [Copyright &y& Elsevier]

In this paper, we study the dynamical behavior of nonautonomous, almost-periodic discrete FitzHugh–Nagumo system defined on infinite lattices. We prove that the nonautonomous infinite-dimensional system has a uniform attractor which attracts all solutions uniformly with respect to the translations of external terms. We also establish the upper semicontinuity of uniform attractors when the infinite-dimensional system is approached by a family of finite-dimensional systems. This paper is based on a uniform tail method, which shows that, for large time, the tails of solutions are uniformly small with respect to bounded initial data as well as the translations of external terms. The uniform tail estimates play a crucial role for proving the uniform asymptotic compactness of the system and the upper semicontinuity of attractors. [ABSTRACT FROM AUTHOR]

Abstract: The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction–diffusion equation has a global attractor in a weighted space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction–diffusion equation is approached by finite-dimensional systems. [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]

This paper deals with the time-dependent Ginzburg–Landau equations of superconductivity in three spatial dimensions. The uniqueness of global weak solutions is established for this model with initial data of the order parameter in L4 and magnetic potential in L3. [Copyright &y& Elsevier]

Deals with the asymptotic behavior of solutions for the Klein-Gordon-Schrodinger equation. Proof for the existence of compact global attractors for the model in the space for each integer; Establishment of the continuity property of the solutions on initial data; Asymptotic compactness of solutions for each integer; Difficulty in obtaining the compactness for weakly dissipative equation.

This paper is concerned with the asymptotic behavior of solutions of the non-autonomous reaction-diffusion equations driven by nonlinear colored noise defined on unbounded domains. The nonlinear drift and diffusion terms are assumed to be continuous but not necessarily Lipschitz continuous which leads to the non-uniqueness of solutions. We prove the existence and uniqueness of pullback random attractors for the multi-valued non-autonomous cocycles generated by the solution operators. The measurability of the random attractors is established by the method based on the weak upper semicontinuity of the solutions. The asymptotic compactness of the solutions is derived by Ball's idea of energy equations in order to overcome the non-compactness of Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stationary approximations of the stochastic wave equations defined on R n with critical exponents. For a class of nonlinear diffusion terms, we prove the existence and uniqueness of tempered pullback attractors of the random wave equations driven by a stationary process as an approximation to the white noise. For a linear multiplicative noise, we prove the upper semicontinuity of these attractors as the step size of the approximations approaches zero. The asymptotic compactness of the solutions on R n is established by the idea of energy equations and the uniform estimates on the tails of the solutions. [ABSTRACT FROM AUTHOR]