scholarly journals Long Term Behavior for a Class of Stochastic Delay Lattice Systems in Xρ Space

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yijin Zhang ◽  
Zongbing Lin
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Additive Noise ◽  
Asymptotic Behavior Of Solutions ◽  
Random Dynamical System ◽  
Lattice Systems ◽  
Stochastic Delay ◽  
Long Term Behavior

In this paper, we focus on the asymptotic behavior of solutions to stochastic delay lattice equations with additive noise and deterministic forcing. We first show the existence of a continuous random dynamical system for the equations. Then we investigate the pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractor in Xρ space. Finally, ergodicity of the systems is achieved.

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

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.

RANDOM ATTRACTORS FOR STOCHASTIC EQUATIONS DRIVEN BY A FRACTIONAL BROWNIAN MOTION

2010 ◽  
Vol 20 (09) ◽  
pp. 2761-2782 ◽  
Author(s):  
M. J. GARRIDO-ATIENZA ◽  
B. MASLOWSKI ◽  
B. SCHMALFUß
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Existence And Uniqueness ◽  
Stochastic Equations ◽  
Hurst Parameter ◽  
Random Dynamical System ◽  
Random Attractors

In this paper, the asymptotic behavior of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/2 is studied. In particular, it is shown that the corresponding solutions generate a random dynamical system for which the existence and uniqueness of a random attractor is proved.

scholarly journals Asymptotic behavior of solutions for a semibounded nonmonotone evolution equation

2003 ◽  
Vol 2003 (9) ◽  
pp. 521-538
Author(s):  
Nikos Karachalios ◽  
Nikos Stavrakakis ◽  
Pavlos Xanthopoulos
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Parabolic Equation ◽  
Evolution Equation ◽  
Existence And Uniqueness ◽  
Nonlinear Parabolic Equation ◽  
Evolution Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Uniqueness Of Solutions ◽  
Nonlinear Parabolic

We consider a nonlinear parabolic equation involving nonmonotone diffusion. Existence and uniqueness of solutions are obtained, employing methods for semibounded evolution equations. Also shown is the existence of a global attractor for the corresponding dynamical system.

scholarly journals Asymptotic Behavior of Stochastic Partly Dissipative Lattice Systems in Weighted Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-23
Author(s):  
Xiaoying Han
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Weighted Space ◽  
Random Dynamical System ◽  
Absorbing Set ◽  
Lattice Systems ◽  
Lattice Differential Equations ◽  
Additive White Noise ◽  
Bounded Absorbing Set ◽  
Infinite Sequences

We study stochastic partly dissipative lattice systems with random coupled coefficients and multiplicative/additive white noise in a weighted space of infinite sequences. We first show that these stochastic partly dissipative lattice differential equations generate a random dynamical system. We then establish the existence of a tempered random bounded absorbing set and a global compact random attractor for the associated random dynamical system.

scholarly journals Random attractors for the stochastic coupled suspension bridge equations of Kirchhoff type

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ling Xu ◽  
Jianhua Huang ◽  
Qiaozhen Ma
Keyword(s):  
Existence And Uniqueness ◽  
Dynamical Behavior ◽  
Suspension Bridge ◽  
Random Dynamical System ◽  
Random Set ◽  
Kirchhoff Type ◽  
Random Attractors ◽  
Long Term Behavior ◽  
Uniqueness Of A Solution

Abstract This paper is devoted to the dynamical behavior of stochastic coupled suspension bridge equations of Kirchhoff type. For the deterministic cases, there are many classical results such as existence and uniqueness of a solution and long-term behavior of solutions. To the best of our knowledge, the existence of random attractors for the stochastic coupled suspension bridge equations of Kirchhoff type is not yet considered. We intend to investigate these problems. We first obtain the dissipativeness of a solution in higher-energy spaces $H^{3}(U)\times H_{0}^{1}(U)\times (H^{2}(U)\cap H_{0}^{1}(U))\times H_{0}^{1}(U)$ H 3 ( U ) × H 0 1 ( U ) × ( H 2 ( U ) ∩ H 0 1 ( U ) ) × H 0 1 ( U ) . This implies that the random dynamical system generated by the equation has a random attractor in $(H^{2}(U)\cap H_{0}^{1}(U))\times L^{2}(U) \times H_{0}^{1}(U)\times L^{2}(U)$ ( H 2 ( U ) ∩ H 0 1 ( U ) ) × L 2 ( U ) × H 0 1 ( U ) × L 2 ( U ) , which is a tempered random set in the space in $H^{3}(U)\times H_{0}^{1}(U)\times (H^{2}(U)\cap H_{0} ^{1}(U))\times H_{0}^{1}(U)$ H 3 ( U ) × H 0 1 ( U ) × ( H 2 ( U ) ∩ H 0 1 ( U ) ) × H 0 1 ( U ) .

RANDOM DIFFERENTIAL EQUATIONS WITH RANDOM DELAYS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 369-388 ◽  
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.

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

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.

scholarly journals Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains

2019 ◽  
Vol 17 (1) ◽  
pp. 1281-1302 ◽  
Author(s):  
Xiaobin Yao ◽  
Xilan Liu
Keyword(s):  
Asymptotic Behavior ◽  
Additive Noise ◽  
Upper Semicontinuity ◽  
Unbounded Domains ◽  
Asymptotic Behavior Of Solutions ◽  
Plate Equation ◽  
Uniform Estimates ◽  
Estimates Of Solutions ◽  
Random Attractors

Abstract We study the asymptotic behavior of solutions to the non-autonomous stochastic plate equation driven by additive noise defined on unbounded domains. We first prove the uniform estimates of solutions, and then establish the existence and upper semicontinuity of random attractors.

Long-term behavior of positive solutions of an exponentially self-regulating system of difference equations

2017 ◽  
Vol 10 (03) ◽  
pp. 1750045 ◽  
Author(s):  
N. Psarros ◽  
G. Papaschinopoulos ◽  
K. B. Papadopoulos
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Difference Equations ◽  
Initial Conditions ◽  
System Of Difference Equations ◽  
Long Term Behavior

In this paper, we study the asymptotic behavior of the positive solutions of a system of the following difference equations: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants and the initial conditions [Formula: see text] and [Formula: see text] are positive numbers.

A note on the generation of random dynamical systems from fractional stochastic delay differential equations

2015 ◽  
Vol 15 (03) ◽  
pp. 1550018 ◽  
Author(s):  
Luu Hoang Duc ◽  
Björn Schmalfuß ◽  
Stefan Siegmund
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Dynamical Systems ◽  
Delay Differential Equations ◽  
Regularity Conditions ◽  
Random Dynamical System ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Stochastic Delay Differential Equation ◽  
Delay Differential

In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a Hölder space which is separable.

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