Uniform attractors for non-autonomous random dynamical systems

Existence of a random attractor for non-autonomous stochastic plate equations with additive noise and nonlinear damping on $\mathbb{R}^{n}$

“…Wang in [25] further extended the concept of asymptotic compactness to the case of partial differential equations with both random and time-dependent forcing terms; moreover, he applied these criteria into the stochastic reaction-diffusion equation with additive noise on R n and obtained the existence of a unique pullback attractor. For most of works on stochastic PDEs, please refer to [9,22,[27][28][29]32] and the references therein.…”

Random attractors for a class of stochastic reaction‐diffusion equations on with multiplicative noise

“…This completes the proof. (14) and (3)- (6) hold. Then, for any v −t ∈ D( − t, ), where D = {D( , ) ∶ ∈ R, ∈ Ω} ∈ , and for any > 0 and P-a.e.…”

“…Theorem 3. Assume that g(x, t) ∈ L 2 loc (R, L 2 (R N )) satisfies (14) and (3)- (6) hold. Then, the RDS associated with problem (1)-(2) has a unique -pullback attractor  ∈  in L 2 (R N ).…”

“…When replacing stochastic term of (1) with additive noise, the existence of random attractor is established in the work of Jiang et al 28 For other studies on the existence of random attractors for stochastic partial differential equations, we refer readers to previous works 6,8,9,20 for details. This paper is organized as follows.…”

Convergences of asymptotically autonomous pullback attractors towards semigroup attractors

“…In dynamical system theory, such a non-autonomous evolution system is often formulated as a process, i.e., a mapping U : R 2 × X → X satisfying U (τ, τ, x) = x and U (t, s, U (s, τ, x)) = U (t, τ, x) for all t s τ and x ∈ X, where R 2 := {(t, τ ) ∈ R 2 : t τ } and X a complete metric space, while the pullback attractor A of a process U is defined as a compact non-autonomous set in the form A = {A(t)} t∈R which is the minimal among those that are invariant and pullback attract non-empty bounded sets in X. The pullback attractor gives rich information of the asymptotic dynamics of the system from the past, and has close relationship to other kinds of attractors, such as uniform attractors, cocycle attractors, etc., see [2,5,3]. Its time-dependence is directly related to the non-autonomous characteristic of the system.…”