scholarly journals Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$

2011 ◽  
Vol 363 (07) ◽  
pp. 3639-3639 ◽  
Author(s):  
Bixiang Wang
Keyword(s):  
Asymptotic Behavior ◽  
Critical Exponents ◽  
Wave Equations ◽  
Stochastic Wave Equations

scholarly journals Multi-valued random dynamics of stochastic wave equations with infinite delays

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jingyu Wang ◽  
Yejuan Wang ◽  
Tomás Caraballo
Keyword(s):  
Asymptotic Behavior ◽  
Wave Equations ◽  
Growth Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Pullback Attractor ◽  
Lipschitz Continuous ◽  
Infinite Delays ◽  
Stochastic Wave Equations ◽  
Additive White Noise ◽  
Stochastic Wave Equation

<p style='text-indent:20px;'>This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.</p>

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>

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