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

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2021-12, Vol.151 (6), p.1700-1730
• Main Authors: Caraballo, Tomás, Guo, Boling, Tuan, Nguyen Huy, Wang, Renhai
• Format: Article
• Language: English
• Subjects: Applied physics; Asymptotic properties; Computational mathematics; Mathematical analysis; Noise; Nonlinearity; Robustness; Time dependence; Viscosity; Wave dispersion; Wave equations
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/prm.2020.77

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).