scholarly journals Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

2018 ◽  
Vol 38 (9) ◽  
pp. 4767-4817 ◽  
Author(s):  
Zhaojuan Wang ◽  
◽  
Shengfan Zhou ◽  
Keyword(s):  
Wave Equation ◽  
White Noise ◽  
Random Attractor ◽  
Exponential Attractor ◽  
Multiplicative White Noise

Random exponential attractor for second-order nonautonomous stochastic lattice systems with multiplicative white noise

2019 ◽  
Vol 19 (06) ◽  
pp. 1950044
Author(s):  
Haijuan Su ◽  
Shengfan Zhou ◽  
Luyao Wu
Keyword(s):  
White Noise ◽  
Weighted Space ◽  
Sufficient Conditions ◽  
Second Order ◽  
Lattice System ◽  
Exponential Attractor ◽  
Lattice Systems ◽  
Random System ◽  
Multiplicative White Noise ◽  
Infinite Sequences

We studied the existence of a random exponential attractor in the weighted space of infinite sequences for second-order nonautonomous stochastic lattice system with linear multiplicative white noise. Firstly, we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle defined on a weighted space of infinite sequences. Secondly, we transferred the second-order stochastic lattice system with multiplicative white noise into a random lattice system without noise through the Ornstein–Uhlenbeck process, whose solutions generate a continuous cocycle on a weighted space of infinite sequences. Thirdly, we estimated the bound and tail of solutions for the random system. Fourthly, we verified the Lipschitz continuity of the continuous cocycle and decomposed the difference between two solutions into a sum of two parts, and carefully estimated the bound of the norm of each part and the expectations of some random variables. Finally, we obtained the existence of a random exponential attractor for the considered system.

scholarly journals Random Attractors for Stochastic Three-Component Reversible Gray-Scott System on Infinite Lattices

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Anhui Gu ◽  
Zhaojuan Wang ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Random Attractors ◽  
Multiplicative White Noise ◽  
Infinite Lattices

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.

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>

scholarly journals Random Attractors for Stochastic Three-Component Reversible Gray-Scott System with Multiplicative White Noise

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Anhui Gu
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Random Attractors ◽  
Multiplicative White Noise

The paper is devoted to proving the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with multiplicative white noise.

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