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

2015 ◽  
Vol 258 (1) ◽  
pp. 148-190 ◽  
Author(s):  
Hongyan Li ◽  
Yuncheng You ◽  
Junyi Tu
Keyword(s):  
Wave Equations ◽  
Nonlinear Damping ◽  
Stochastic Wave Equations ◽  
Random Attractors

scholarly journals Random attractors for stochastic delay wave equations on $ \mathbb{R}^n $ with linear memory and nonlinear damping

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jingyu Wang ◽  
Yejuan Wang ◽  
Lin Yang ◽  
Tomás Caraballo
Keyword(s):  
Wave Equation ◽  
White Noise ◽  
Unbounded Domain ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Nonlinear Damping ◽  
Random Attractor ◽  
Stochastic Delay ◽  
Additive White Noise ◽  
Random Attractors

<p style='text-indent:20px;'>A non-autonomous stochastic delay wave equation with linear memory and nonlinear damping driven by additive white noise is considered on the unbounded domain <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. We establish the existence and uniqueness of a random attractor <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> that is compact in <inline-formula><tex-math id="M3">\begin{document}$ C{([-h, 0];H^1(\mathbb{R}^n))}\times C{([-h, 0];L^2(\mathbb{R}^n))}\times L_\mu^2(\mathbb{R}^+;H^1(\mathbb{R}^n)) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ 1\leqslant n \leqslant 3 $\end{document}</tex-math></inline-formula>.</p>

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

2020 ◽  
pp. 1-31
Author(s):  
Tomás Caraballo ◽  
Boling Guo ◽  
Nguyen Huy Tuan ◽  
Renhai Wang
Keyword(s):  
Wave Equations ◽  
Unbounded Domains ◽  
Nonlinear Wave Equations ◽  
Entire Space ◽  
Independent Function ◽  
Decomposition Approach ◽  
Stochastic Wave Equations ◽  
Operator Type ◽  
Random Attractors ◽  
Dissipative Terms

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).

scholarly journals STOCHASTIC WAVE EQUATIONS WITH NONLINEAR DAMPING AND SOURCE TERMS

2013 ◽  
Vol 16 (02) ◽  
pp. 1350013 ◽  
Author(s):  
HONGJUN GAO ◽  
FEI LIANG ◽  
BOLING GUO
Keyword(s):  
Wave Equations ◽  
Blow Up ◽  
Energy Inequality ◽  
Local Existence ◽  
Galerkin Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Damping ◽  
Local Existence And Uniqueness ◽  
Stochastic Wave Equations

In this paper, we discuss an initial boundary value problem for the stochastic wave equation involving the nonlinear damping term |ut|q–2utand a source term of the type|u|p–2u. We firstly establish the local existence and uniqueness of solution by the Galerkin approximation method and show that the solution is global for q ≥ p. Secondly, by an appropriate energy inequality, the local solution of the stochastic equations will blow up with positive probability or explosive in energy sense for p > q.

RANDOM ATTRACTORS FOR DAMPED STOCHASTIC WAVE EQUATIONS WITH MULTIPLICATIVE NOISE

2008 ◽  
Vol 19 (04) ◽  
pp. 421-437 ◽  
Author(s):  
XIAOMING FAN
Keyword(s):  
Fractal Dimension ◽  
Dynamical Systems ◽  
Wave Equations ◽  
Multiplicative Noise ◽  
Random Dynamical Systems ◽  
Stochastic Wave Equations ◽  
Random Attractors

In this paper, we investigate the existence of compact random attractors and their fractal dimension for the random dynamical systems determined by damped stochastic wave equations of Sine–Gordon type with linear multiplicative noise.

scholarly journals Asymptotic behavior of supercritical wave equations driven by colored noise on unbounded domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bixiang Wang
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Colored Noise ◽  
Unbounded Domains ◽  
Strichartz Estimates ◽  
Well Posedness ◽  
Natural Energy ◽  
Random Attractors ◽  
Supercritical Nonlinearity

<p style='text-indent:20px;'>This paper deals with the asymptotic behavior of the non-autonomous random dynamical systems generated by the wave equations with supercritical nonlinearity driven by colored noise defined on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ n\le 6 $\end{document}</tex-math></inline-formula>. Based on the uniform Strichartz estimates, we prove the well-posedness of the equation in the natural energy space and define a continuous cocycle associated with the solution operator. We also establish the existence and uniqueness of tempered random attractors of the equation by showing the uniform smallness of the tails of the solutions outside a bounded domain in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.</p>

Sign in / Sign up

Export Citation Format

Share Document