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Register a new userExistence and Upper Semicontinuity of Attractors for Stochastic Equations with Deterministic Non-autonomous Terms
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Bixiang Wang
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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.
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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.
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.
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 article we study the asymptotic behavior of solutions, in sense of global pullback attractors, of the evolution system $$ begin{cases} u_{tt} +etaDelta^2 u+a(t)Deltatheta=f(t,u), & t>tau, xinOmega, theta_t-kappaDelta theta-a(t)Delta u_t=0, & t>tau, xinOmega, end{cases} $$ subject to boundary conditions $$ u=Delta u=theta=0, t>tau, xinpartialOmega, $$ where $Omega$ is a bounded domain in $mathbb{R}^N$ with $Ngeqslant 2$, which boundary $partialOmega$ is assumed to be a $mathcal{C}^4$-hypersurface, $eta>0$ and $kappa>0$ are constants, $a$ is an Holder continuous function, $f$ is a dissipative nonlinearity locally Lipschitz in the second variable.
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.
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