Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

Abstract

We study some time-related properties of the random attractor for the stochastic wave equation on an unbounded domain with time-varying coefficient and force. We assume that the coefficient is bounded and the timedependent force is backward tempered, backward complement-small, backward tail-small, and then prove both existence and backward compactness of a random attractor on the universe of all backward tempered sets. By using the Egoroff and Lusin theorems, we show the measurability of the absorbing set although it is the union of some random sets over an uncountable index set. Moreover, we obtain the backward compactness of the attractor if the force is periodic, and obtain the periodicity of the attractor if both force and coefficient are periodic.

Random attractors for stochastic time-dependent damped wave equation with critical exponents

“…In [16], the compactness for deterministic equation formulated as (1.1) was obtained from taking advantages of asymptotically smoothness and Lyapunov functions. For equations similar to our case, in [8,30], the existence of random attractor was proved but with subcritical nonlinearity. As for our critical case, we apply the energy method in [3] to overcome the lack of compactness, which proves the asymptotic compactness of the system.…”

“…where ζ 1 , ζ 2 are positive constants(see [28,29,30,31,33,34] and references therein). During the process for proving a priori estimate there would usually be a product of the nonlinearity and the random term, namely (f (u), µ(θ t ω)) L 2 .…”

Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains

“…We remark that the time-semi-uniform compactness of non-autonomous attractors and kernel sections has been recently investigated in [30,32,55,56] and [43,44] for deterministic and stochastic PDEs, respectively. The asymptotically autonomous robustness of non-autonomous attractors was studied for deterministic equations [24-26, 31, 45].…”

Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory

“…Robustness of delayed attractors for partly convergent cocycles. In this section, we first recall the general concept of a pullback random attractor which is first introduced by Wang [33] with developments in [1,9,10,25,36,47], and then establish a new robustness theorem of delayed PRAs to reduce the convergence condition of cocycles in [39, theorem 2.1].…”