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

<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>

2019 ◽  
Vol 20 (03) ◽  
pp. 2050018
Author(s):  
Lin Shi ◽  
Dingshi Li ◽  
Xiliang Li ◽  
Xiaohu Wang

We investigate the asymptotic behavior of a class of non-autonomous stochastic FitzHugh–Nagumo systems driven by additive white noise on unbounded thin domains. For this aim, we first show the existence and uniqueness of random attractors for the considered equations and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain.


Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yanfeng Guo ◽  
Chunxiao Guo ◽  
Yongping Xi

Some dynamics behaviors for the nonautonomous stochastic fifth-order Swift–Hohenberg equation with additive white noise are considered. The existence of pullback random attractors for the nonautonomous stochastic fifth-order Swift–Hohenberg equation with some properties is mainly investigated on the bounded domain and unbounded domain, through the Ornstein–Uhlenbeck transformation and tail-term estimates. Furthermore, on the basis of some conditions, the finiteness of fractal dimension of random attractor is proved.


2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Xiaoquan Ding ◽  
Jifa Jiang

This paper is devoted to a stochastic retarded lattice dynamical system with additive white noise. We extend the method of tail estimates to stochastic retarded lattice dynamical systems and prove the existence of a compact global random attractor within the set of tempered random bounded sets.


2009 ◽  
Vol 09 (02) ◽  
pp. 293-313 ◽  
Author(s):  
HONGJUN GAO ◽  
CHENGFENG SUN

In this article, we obtain the existence and uniqueness of strong solutions to 3D viscous stochastic primitive equations (PEs) and the random attractor for 3D viscous PEs with additive white noise.


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bixiang Wang

<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>


2019 ◽  
Vol 17 (1) ◽  
pp. 89-103
Author(s):  
Qiaozhen Ma ◽  
Jing Wang ◽  
Tingting Liu

Abstract In this article, we consider the long-time behavior of solutions for the wave equation with nonlinear damping and linear memory. Within the theory of process on time-dependent spaces, we verify the process is asymptotically compact by using the contractive functions method, and then obtain the existence of the time-dependent attractor in $\begin{array}{} H^{1}_0({\it\Omega})\times L^{2}({\it\Omega})\times L^{2}_{\mu}(\mathbb{R}^{+};H^{1}_0({\it\Omega})) \end{array}$.


2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Anhui Gu ◽  
Zhaojuan Wang ◽  
Shengfan Zhou

We prove the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices.


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