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

Abstract

This paper is concerned with the asymptotic behaviour of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space $\mathbb {R}^n$ . The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustness of pullback random attractors are proved in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ when the growth rate of the nonlinearity has a subcritical range, the density of the noise is suitably controllable, and the time-dependent force converges to a time-independent function in some sense. The main difficulty to establish the time-semi-uniform pullback asymptotic compactness of the solutions in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ is caused by the lack of compact Sobolev embeddings on $\mathbb {R}^n$ , as well as the weak dissipativeness of the equations is surmounted at light of the idea of uniform tail-estimates and a spectral decomposition approach. The measurability of random attractors is proved by using an argument which considers two attracting universes developed by Wang and Li (Phys. D 382: 46–57, 2018).

Asymptotic Autonomy of Random Attractors in Regular Spaces for Non-autonomous Stochastic Navier-Stokes Equations

“…For example, one can consider f (x, t) = (e t + 1)f 0 (x) and f ∞ = f 0 (x) and they satisfy the above Hypothesis 3.1 (see [6]). The following Lemma is adapted from the work [33].…”

“…Moreover, the authors in [33] (Section 5) demonstrated abstract results for asymptotic autonomy of random attractors. For asymptotic autonomy of random attractors, we refer the readers to [6,33,53], etc. To the best of our knowledge, there is no result available in the literature on the existence of backward compact random attractors as well as for their asymptotic autonomy for 2D SNSE driven by multiplicative noise.…”

On a time fractional diffusion with nonlocal in time conditions

“…Time-fractional partial differential equations are well known to describe modeling of anomalously slow transport processes. These models are often expressed in the form of fractional diffusion or subdiffusion equations which have many applications in various kinds of research areas, e.g., thermal diffusion in fractal domains [1] and protein dynamics [2], we can refer for more details to [3][4][5][6]. By replacing many differential operators of fractional order with different type of PDEs of integer order, we formulate various types of boundary value problems with fractional order.…”

Existence and approximation of attractors for nonlinear coupled lattice wave equations

“…It is well known that attractors play a key role in the study of long-time behavior of dissipative or weakly dissipative evolution equations. In recent years, the attractors theory and application have been well-developed in the literature for many kinds equations, see e.g., [5,4,9,7,12,11,10,14,8,26,27,30,29,28,35,38,39,40,41,42,43,44,48] and references therein. In addition, the existence and properties of attractors of LDEs have been extensively investigated in many publications [1,2,3,19,22,18,17,20,21,24,23,16,36,55,56,57,58].…”