Existence of Random Attractors for ap-Laplacian-Type Equation with Additive Noise

Abstract and Applied Analysis ◽

10.1155/2011/616451 ◽

2011 ◽

Vol 2011 ◽

pp. 1-21 ◽

Cited By ~ 7

Author(s):

Wenqiang Zhao ◽

Yangrong Li

Keyword(s):

Dynamical System ◽

Additive Noise ◽

Single Point ◽

Type Equation ◽

Random Attractor ◽

Approximation Procedure ◽

Restrictive Assumption ◽

Stochastic Dynamical System ◽

Additive White Noise ◽

Uniqueness Of A Solution

We first establish the existence and uniqueness of a solution for a stochasticp-Laplacian-type equation with additive white noise and show that the unique solution generates a stochastic dynamical system. By using the Dirichlet forms of Laplacian and an approximation procedure, the nonlinear obstacle, arising from the additive noise is overcome when we make energy estimate. Then, we obtain a random attractor for this stochastic dynamical system. Finally, under a restrictive assumption on the monotonicity coefficient, we find that the random attractor consists of a single point, and therefore the system possesses a unique stationary solution.

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Random Attractors for Stochastic Retarded Lattice Dynamical Systems

Abstract and Applied Analysis ◽

10.1155/2012/409282 ◽

2012 ◽

Vol 2012 ◽

pp. 1-27 ◽

Cited By ~ 4

Author(s):

Xiaoquan Ding ◽

Jifa Jiang

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

White Noise ◽

Random Attractor ◽

Additive White Noise ◽

Lattice Dynamical System ◽

Random Attractors ◽

Lattice Dynamical Systems ◽

Bounded Sets

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.

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RANDOM ATTRACTOR FOR THE 3D VISCOUS STOCHASTIC PRIMITIVE EQUATIONS WITH ADDITIVE NOISE

Stochastics and Dynamics ◽

10.1142/s0219493709002683 ◽

2009 ◽

Vol 09 (02) ◽

pp. 293-313 ◽

Cited By ~ 3

Author(s):

HONGJUN GAO ◽

CHENGFENG SUN

Keyword(s):

White Noise ◽

Existence And Uniqueness ◽

Additive Noise ◽

Strong Solutions ◽

Random Attractor ◽

Primitive Equations ◽

Additive White Noise

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.

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RANDOM DIFFERENTIAL EQUATIONS WITH RANDOM DELAYS

Stochastics and Dynamics ◽

10.1142/s0219493711003358 ◽

2011 ◽

Vol 11 (02n03) ◽

pp. 369-388 ◽

Cited By ~ 30

Author(s):

M. J. GARRIDO-ATIENZA ◽

A. OGROWSKY ◽

B. SCHMALFUSS

Keyword(s):

Dynamical System ◽

Differential Equation ◽

Existence And Uniqueness ◽

Existence Result ◽

Random Dynamical System ◽

Random Delay ◽

Random Differential Equation ◽

Autonomous Case ◽

Random Differential Equations ◽

Uniqueness Of A Solution

We investigate a random differential equation with random delay. First the non-autonomous case is considered. We show the existence and uniqueness of a solution that generates a cocycle. In particular, the existence of an attractor is proved. Secondly we look at the random case. We pay special attention to the measurability. This allows us to prove that the solution to the random differential equation generates a random dynamical system. The existence result of the attractor can be carried over to the random case.

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Nonlinear Stochastic Dynamical System of Bio-Dissimilation of Glycerol to 1,3-Propanediol in Batch Culture and Its Viable Set

2009 3rd International Conference on Bioinformatics and Biomedical Engineering ◽

10.1109/icbbe.2009.5163119 ◽

2009 ◽

Cited By ~ 1

Author(s):

Lei Wang ◽

Enmin Feng ◽

Zhilong Xiu

Keyword(s):

Dynamical System ◽

Batch Culture ◽

Stochastic Dynamical System

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Random attractor for the stochastic Cahn–Hilliard–Navier–Stokes system with small additive noise

Stochastic Analysis and Applications ◽

10.1080/07362994.2018.1431130 ◽

2018 ◽

Vol 36 (3) ◽

pp. 546-559 ◽

Cited By ~ 3

Author(s):

Fang Li ◽

Bo You

Keyword(s):

Stokes System ◽

Additive Noise ◽

Random Attractor ◽

Navier Stokes ◽

Small Additive

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Derivative-Free Observability Analysis of a Stochastic Dynamical System

IEEE Transactions on Network Science and Engineering ◽

10.1109/tnse.2021.3093535 ◽

2021 ◽

pp. 1-1

Author(s):

Zongsheng Zheng ◽

Yijun Xu ◽

Lamine Mili ◽

Zhigang Liu ◽

Long Peng ◽

...

Keyword(s):

Dynamical System ◽

Stochastic Dynamical System ◽

Observability Analysis ◽

Derivative Free

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Asymptotic behavior of non-autonomous fractional p-Laplacian equations driven by additive noise on unbounded domains

Bulletin of Mathematical Sciences ◽

10.1142/s1664360720500204 ◽

2020 ◽

pp. 2050020

Author(s):

Renhai Wang ◽

Bixiang Wang

Keyword(s):

Asymptotic Behavior ◽

Existence And Uniqueness ◽

Additive Noise ◽

Random Coefficients ◽

Asymptotic Behavior Of Solutions ◽

Uniqueness Of Solutions ◽

Additive White Noise ◽

Approach Zero ◽

Drift Terms ◽

Laplacian Equations

This paper deals with the asymptotic behavior of solutions to non-autonomous, fractional, stochastic [Formula: see text]-Laplacian equations driven by additive white noise and random terms defined on the unbounded domain [Formula: see text]. We first prove the existence and uniqueness of solutions for polynomial drift terms of arbitrary order. We then establish the existence and uniqueness of pullback random attractors for the system in [Formula: see text]. This attractor is further proved to be a bi-spatial [Formula: see text]-attractor for any [Formula: see text], which is compact, measurable in [Formula: see text] and attracts all random subsets of [Formula: see text] with respect to the norm of [Formula: see text]. Finally, we show the robustness of these attractors as the intensity of noise and the random coefficients approach zero. The idea of uniform tail-estimates as well as the method of higher-order estimates on difference of solutions are employed to derive the pullback asymptotic compactness of solutions in [Formula: see text] for [Formula: see text] in order to overcome the non-compactness of Sobolev embeddings on [Formula: see text] and the nonlinearity of the fractional [Formula: see text]-Laplace operator.

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Dynamics of stochastic FitzHugh–Nagumo systems with additive noise on unbounded thin domains

Stochastics and Dynamics ◽

10.1142/s0219493720500185 ◽

2019 ◽

Vol 20 (03) ◽

pp. 2050018

Author(s):

Lin Shi ◽

Dingshi Li ◽

Xiliang Li ◽

Xiaohu Wang

Keyword(s):

Asymptotic Behavior ◽

White Noise ◽

Unbounded Domain ◽

Existence And Uniqueness ◽

Additive Noise ◽

Upper Semicontinuity ◽

Thin Domains ◽

Additive White Noise ◽

Random Attractors ◽

Lower Dimensional

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.

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