This paper deals with the multivalued non-autonomous random dynamical system generated by the non-autonomous stochastic wave equations on unbounded domains, which has a non-Lipschitz nonlinearity with critical exponent in the three dimensional case. We introduce the concept of weak upper semicontinuity of multivalued functions and use such continuity to prove the measurability of multivalued functions from a metric space to a separable Banach space. By this approach, we show the measurability of pullback attractors of the multivalued random dynamical system of the wave equations regardless of the completeness of the underlying probability space. The asymptotic compactness of solutions is proved by the method of energy equations, and the difficulty caused by the non-compactness of Sobolev embeddings on $R^n$ is overcome by the uniform estimates on the tails of solutions.
By the Lyapunov-Perron method,we prove the existence of random inertial manifolds for a class of equations driven simultaneously by non-autonomous deterministic and stochastic forcing. These invariant manifolds contain tempered pullback random attractors if such attractors exist. We also prove pathwise periodicity and almost periodicity of inertial manifolds when non-autonomous deterministic forcing is periodic and almost periodic in time, respectively.
In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by non-autonomous stochastic equations. These solutions are pathwise stochastic analogues of deterministic dynamical systems. The existence and bifurcation of random periodic (random almost periodic, random almost automorphic) solutions have been established for a one-dimensional stochastic equation with multiplicative noise.
This paper is concerned with pullback attractors of the stochastic p-Laplace equation defined on the entire space R^n. We first establish the asymptotic compactness of the equation in L^2(R^n) 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 R^n is overcome by the uniform smallness of solutions outside a bounded domain.
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.
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.
This paper is concerned with the asymptotic behavior of solutions of the two-dimensional Navier-Stokes equations with both non-autonomous deterministic and stochastic terms defined on unbounded domains. We first introduce a continuous cocycle for the equations and then prove the existence and uniqueness of tempered random attractors. We also characterize the structures of the random attractors by complete solutions. When deterministic forcing terms are periodic, we show that the tempered random attractors are also periodic. Since the Sobolev embeddings on unbounded domains are not compact, we establish the pullback asymptotic compactness of solutions by Balls idea of energy equations.
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. As an application of the abstract theory, we prove the existence of a unique pullback attractor for Reaction-Diffusion equations on $R^n$ 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.
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.
The upper semicontinuity of random attractors for non-compact random dynamical systems is proved when the union of all perturbed random attractors is precompact with probability one. This result is applied to the stochastic Reaction-Diffusion with white noise defined on the entire space R^n.
