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

Part of: Infinite-dimensional dissipative dynamical systems Partial differential equations Topological dynamics

Published online by Cambridge University Press:  05 November 2020

Tomás Caraballo ,
Boling Guo ,
Nguyen Huy Tuan  and
Renhai Wang
Tomás Caraballo
Affiliation:
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla, Spain
Boling Guo
Affiliation:
Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing100088, China
Nguyen Huy Tuan
Affiliation:
Department of Mathematics and Computer Science, University of Science-VNUHCM, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Viet Nam
Renhai Wang*
Affiliation:
Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing100088, China ([email protected])
*
*Corresponding author.
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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).

Keywords

Weakly dissipative wave equationpullback random attractorsasymptotically autonomous robustnesstime-semi-uniform compactnessoperator-type noise

MSC classification

Primary: 37L55: Infinite-dimensional random dynamical systems; stochastic equations
Secondary: 37B55: Nonautonomous dynamical systems 35B41: Attractors 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 1700 - 1730
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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