In this paper, we establish the existence of random attractors for stochastic parabolic equations driven by additive noise as well as deterministic non-autonomous forcing terms in weighted Lebesgue spaces $ L_{\delta}^r(\mathcal{O})$, where $ 1<r<\infty ,\ \delta$ is the distance from $ x$ to the boundary. The nonlinearity $ f(x,u)$ of equation depending on the spatial variable does not have the bound on the derivative in $ u$, and then causes critical exponent. In both subcritical and critical cases, we get the well-posedness and dissipativeness of the problem under consideration and, by smoothing property of heat semigroup in weighted space, the asymptotical compactness of random dynamical system corresponding to the original system.

This paper investigates the stochastic strongly damped wave equation with critical nonlinearity driven by a spatially homogeneous Winer process on the whole space. We first establish the existence of martingale solutions for the equation. Moreover, we prove that the martingale solution of the equation with deterministic initial data is pathwise unique and obtain a pathwise unique strong solution in the sense of probability. Finally, we construct an invariant measure for the equation with critical nonlinearity by using a weak Feller approach and construct an invariant measure for the equation with subcritical nonlinearity by using a decomposition method. We mention that the invariant measure constructed by the decomposition method is more regular than that constructed by the weak Feller approach.

This paper investigates the long-time behavior of a stochastic strongly damped wave equation with additive noise on $\mathbb{R}$^N. We establish that there exists a unique pullback random attractor for the equation in natural space $\mathcal{H}=H^1(\mathbb{R}^{N})\times L^2(\mathbb{R}^{N})$ with the nonlinearity g(x,u) being of optimal subcritical growth p: 1≤p<p^* ≡ N+2/(N-2)(N ≥ 3). In addition, we get the upper semicontinuity of the pullback random attractor as the intensity of noise goes to zero.