Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping

Abstract

This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with nonlinear damping and multiplicative white noise defined on an unbounded domain. By showing the pullback asymptotic compactness of the cocycle in a certain parameter region, we prove the existence of a random attractor when the intensity of noise is sufficiently small. For the stochastic wave equation with rapidly oscillating external force we prove that the Hausdorff distance between the random attractor A of the original equation and the random attractor A 0 of the averaged equation is in the order of O( ).

Probabilistic Robustness for Dispersive-Dissipative Wave Equations Driven by Small Laplace-Multiplier Noise

“…A random compact set A ε ∈ D is said to be a D-random attractor for the RDS Φ ε (given by (18)) if it is invariant, i.e. Φ ε (t, ω)A ε (ω) = A ε (θ t ω) for t ≥ 0, ω ∈ Ω 0 , and…”

“…The wave equation without the dispersive term was also discussed in Wang [21] and Yang, Duan and Kloeden [24] for such additive noise and in Wang, Zhou and Gu [22] for usual multiplicative noise, i.e. Su = u, also see [8,9,11,13,17,18,20,25,26,32].…”

Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

“…The random attractor and the bounds of its Hausdorff and fractal dimensions for the stochastic wave equations with additive noise (i.e., the random term in (1) is "adW (t)" independent of u) have been studied by many authors, see [12,13,8,18,21,38,54,57,63,65]. For the stochastic system (1) with linear multiplicative noise "au • dW (t)" (depending on the state variable u) and sufficient small coefficient |a| of random term, when the nonlinear function f has a subcubic growth exponent (i.e., f 1 ≡ 0 in (A1)), the existence and the boundedness of fractal dimension of random attractor were studied, see [22,36,52,66], of those, Zhou and Zhao in [66] gave some sufficient conditions to bound the fractal dimension of a random invariant set for a cocycle and applied these conditions to get an upper bound of fractal dimension of the random attractor of system (1).…”

“…In this case, there are two main essential difficulties. The first difficulty arises in showing the asymptotic compactness of system that is the key step to prove the existence of a random attractor, which is caused by the cubic growth condition (2) of f and can not be overcome by decomposing the solutions of system just one time like in the deterministic case [27,58,62] or the subcubic growth exponential case [36,52,65,66]. The second difficulty occurs in the possible unboundedness of the expectation of bound of random attractor in a "higher regularity" space here, which is a basic requiring condition in known existing methods to show the boundedness of the fractal dimension of a random attractor [34,56,65,66].…”

Random attractor for stochastic non-autonomous damped wave equation with critical exponent

“…For the non-autonomous stochastic wave system (1) on the unbounded domain, when f has a subcritical growth exponent, Wang and Li et al proved the existence of a random attractor, see [16,20,22].…”