Non-autonomous Stochastic Reaction-diffusion Equations Research Articles

Pullback measure attractors for non-autonomous stochastic reaction-diffusion equations on thin domains

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.

Asymptotical behavior of non-autonomous stochastic reaction–diffusion equations with variable delay on $${\mathbb {R}}^N$$

Asymptotical behavior of non-autonomous stochastic reaction–diffusion equations with variable delay on $${\mathbb {R}}^N$$

Invariant measures and stochastic Liouville type theorem for non-autonomous stochastic reaction-diffusion equations

This paper is concerned with the invariant measures of non-autonomous stochastic reaction-diffusion equations on unbounded domains. We firstly investigate the existence of invariant measures for random dynamical systems over two parametric spaces as well as periodic invariant measures, and then discuss the convergence of (periodic) invariant measures with respect to some system parameters. Random Liouville type equation is also derived for invariant measures associated with non-autonomous random differential equations. Based on such abstract results, we prove the existence of (periodic) invariant measures for non-autonomous stochastic reaction-diffusion equation on unbounded domains, which are supported on the pullback attractors. And we further show every limit of a sequence of (periodic) invariant measures must be the (periodic) invariant measure of the corresponding limiting equation when the noise intensity varies in finite interval. Moreover, we prove that the invariant measures of the underlying equation satisfy a stochastic Liouville type equation.

Regular random attractors for non-autonomous stochastic evolution equations with time-varying delays on thin domains

This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction–diffusion equations with time-varying delays on thin domains. First, we prove the existence and uniqueness of the regular random attractor. Then, we prove the upper semicontinuity of the regular random attractors for the equations on a family of (n + 1)-dimensional thin domains that collapses onto an n-dimensional domain.

Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains

<p style='text-indent:20px;'>This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction-diffusion equations on thin domains. Firstly, we prove the existence and uniqueness of the regular random attractor. Then we prove the upper semicontinuity of the regular random attractors for the equations on a family of <inline-formula><tex-math id="M1">$ (n+1) $</tex-math></inline-formula>-dimensional thin domains collapses onto an <inline-formula><tex-math id="M2">$ n $</tex-math></inline-formula>-dimensional domain.</p>

Upper semi-continuity of random attractors for a non-autonomous dynamical system with a weak convergence condition

In this paper, we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit. As an application, we obtain the convergence of random attractors for non-autonomous stochastic reaction-diffusion equations on unbounded domains, when the density of stochastic noises approaches zero. The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem. A differentiability condition on nonlinearity is omitted, which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity. These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.

Random dynamics for non-autonomous stochastic evolution equations without uniqueness on unbounded narrow domains

This paper deals with the limiting behavior of non-autonomous stochastic reaction-diffusion equations without uniqueness on unbounded narrow domains. We prove the existence and upper semicontinuity of random attractors for the equations on a family of unbounded (n + 1)-dimensional narrow domains, which collapses onto an n-dimensional domain. Since the solutions are non-uniqueness, which leads to a multivalued random dynamical system with the solution operators of the equation, we will prove the existence and upper semicontinuity of attractors by multivalued random dynamical system theory.

Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains

This article is concerned with the limiting behavior of dynamics of a class of non-autonomous stochastic partial differential equations driven by colored noise on unbounded thin domains. We first prove the existence of tempered pullback random attractors for the equations defined on $ (n+1) $-dimensional unbounded thin domains. Then, we show the upper semicontinuity of these attractors when the $ (n+1) $-dimensional unbounded thin domains collapse onto the $ n $-dimensional space $ \mathbb{R}^n $. Here, the tail estimates are utilized to deal with the non-compactness of Sobolev embeddings on unbounded domains.

Limiting behavior of non-autonomous stochastic reaction-diffusion equations on unbounded thin domains

This article is concerned with the limiting behavior of dynamics of a class of nonautonomous stochastic partial differential equations driven by multiplicative white noise on unbounded thin domains. We first prove the existence of tempered pullback random attractors for the equations defined on (n + 1)-dimensional unbounded thin domains. Then, we show the upper semicontinuity of these attractors when the (n + 1)-dimensional unbounded thin domains collapse onto the n-dimensional space Rn. Here, the tail estimates are utilized to deal with the noncompactness of Sobolev embeddings on unbounded domains.

Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$

In this paper we study the asymptotic dynamics of the weak solutions of nonautonomous stochastic reaction-diffusion equations driven by a time-dependent forcing term and the multiplicative noise. By conducting the uniform estimates we show that the cocycle generated by this SRDE has a pullback \begin{document}$(L^2, H^1)$\end{document} absorbing set and it is pullback asymptotically compact through the pullback flattening approach. The existence of a pullback \begin{document}$(L^2, H^1)$\end{document} random attractor for this random dynamical system in space \begin{document}$H^{1}(\mathbb{R}^{n})$\end{document} is proved.

Limiting behavior of non-autonomous stochastic reaction–diffusion equations on thin domains

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.