具有线性乘积白噪声的随机非自治吊桥方程的随机指数吸引子

本文考虑具有线性乘积白噪声的随机非自治吊桥方程长时间行为.首先,建立了所研究共圈系统的适定性;第二步,研究了该系统随机吸引子的存在性;第三步,当随机系数趋于0时,得到了随机吸引子的上半连续性;第四步,通过``迭代''法证明了随机吸引子在高正则空间中的正则性;最后,给出了该系统随机指数吸引子的存在性,同时得到了吸引子的有限分形维数.     

非自治随机FitzHugh-Nagumo系统的随机一致指数吸引子

本文首先给出了非自治随机动力系统的随机一致指数吸引子的概念及其存在性判据,其次证明了Rn上的带加法噪声和拟周期外力的FitzHugh-Nagumo系统的随机一致指数吸引子的存在性.     

带可加噪声的非自治随机Boussinesq格点方程的随机吸引子

该文研究具时变耦合系数的非自治随机Boussinesq格点系统同时受依时间确定性外力和可加白噪声影响时的渐近行为.首先证明非自治随机Boussinesq格点方程的解生成的连续余圈的随机吸引子的存在性.然后证明此系统的随机吸引子在噪声项系数趋于零时的上半连续性.     

强阻尼半线性波动方程的全局吸引子

本文讨论一类带强阻尼项的半线性波动方程的全局吸引子的存在性．首先给出了方程解的存在唯一性定理，建立了解的C°-半群；然后运用Hale提出的a-收缩理论，证明了该类方程存在全局吸引子．     

动态边界上随机波动方程的吸引子

研究了一类动态边界上的随机波动方程.通过建立一种分解技术,证明了方程随机吸引子的存在性.分解同时表明,该吸引子上的点(或者解)一定满足某种稳定的边界条件.最后,证明了吸引子的结构与分解所得的静态边界上波动方程的随机吸引子相同.     

随机反应扩散方程的L~P-随机吸引子

研究了定义在无界区域上具可乘白噪音的随机反应扩散方程的渐近行为.运用一致估计得到了U3-随机吸收集;对方程的解运用渐近优先估计法,建立了相应随机动力系统的渐近紧性,证明了LP-随机吸引子的存在性.该随机吸引子是紧不变集并按LP-范数吸L2中所有缓增集,其中,非线性项/满足p-1(p≥2)阶增长条件.     

带跳的随机Duffing-Van Der Pol方程的随机吸引子(英文)

本文利用随机动力系统和随机分析方法,研究了在一定条件下带跳的随机Duffing-van der Pol方程随机吸引子的存在性和随机分岔.     

无界区域非自治随机sine-Gordon方程的D-周期吸引子

研究非自治随机sine-Gordon方程所生成的随机动力系统φ的D-周期吸引子的存在唯一性.运用一致估计得到了D的D-吸收集的存在性,并用分解技巧,证明了φ的渐近紧性,建立了动力系统φ在H~1(R~n)×L~2(R~n)中的D-周期吸引子的存在唯一性.当非自治外力项具有周期性时,该D-吸引子也呈现相同的周期性.     

一类非自治分数阶随机波动方程的随机吸引子

本文考虑带加性噪声的非自治分数阶随机波动方程在无界区域R~n上的渐近行为.首先将随机偏微分方程转化为随机方程,其解产生一个随机动力系统,然后运用分解技术建立该系统的渐近紧性,最后证明随机吸引子的存在性.     

带阻尼的随机浅水波方程的随机吸引子

研究了在H~1(R)中带阻尼的随机浅水波方程的随机吸引子的存在性.主要工具是Fourier限制范数方法以及将解分解为衰减部分与正则部分.     

Random Point Attractors Versus Random Set Attractors

The notion of an attractor for a random dynamical system withrespect to a general collection of deterministic sets is introduced.This comprises, in particular, global point attractors and globalset attractors. After deriving a necessary and sufficient conditionfor existence of the corresponding attractors it is proved thata global set attractor always contains all unstable sets ofall of its subsets. Then it is shown that in general randompoint attractors, in contrast to deterministic point attractors,do not support all invariant measures of the system. However,for white noise systems it holds that the minimal point attractorsupports all invariant Markov measures of the system.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Reaction-diffusion Equations

This paper dealswith non-autonomous fractional stochastic reaction-diffusion equations driven by multiplicative noise with s ∈ (0,1). We first present some conditions for estimating the boundedness of fractal dimension of a random invariant set. Then we establish the existence and uniqueness of tempered pullback random attractors. Finally, the finiteness of fractal dimension of the random attractors is proved.     

Conceptual Analysis and Random Attractor for Dissipative Random Dynamical Systems

The aim of this work is to understand better the long time behaviour of asymptotically compact random dynamical systems （RDS）, which can be generated by solutions of some stochastic partial differential equations on unbounded domains. The conceptual analysis for the long time behavior of RDS will be done through some examples. An application of those analysis will be demonstrated through the proof of the existence of random attractors for asymptotically compact dissipative RDS.     

Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise

We study the asymptotic behavior of solutions to the stochastic sine-Gordon lattice equations with multiplicative white noise. We first prove the existence and uniqueness of solutions, and then establish the existence of tempered random bounded absorbing sets and global random attractors.     

Random attractors of stochastic non-newtonian fluids

The present paper investigates the asymptotic behavior of solutions for stochastic non-Newtonian fluids in a two-dimensional domain. Firstly, we prove the existence of random attractors ΛH(ω) in H; Secondly, we prove the existence of random attractors ΛV (ω) in V . Then we verify regularity of the random attractors by showing that ΛH(ω) = ΛV (ω), which implies the smoothing effect of the fluids in the sense that solution becomes eventually more regular than the initial data.     

Random attractors for stochastic lattice dynamical systems in weighted spaces

In this paper, we first provide some sufficient conditions for the existence of global compact random attractors for general random dynamical systems in weighted space (p?1) of infinite sequences. Then we consider the existence of global compact random attractors in weighted space for stochastic lattice dynamical systems with random coupled coefficients and multiplicative/additive white noises. Our results recover many existing ones on the existence of global random attractors for stochastic lattice dynamical systems with multiplicative/additive white noises in regular l² space of infinite sequences.     

Attractors for a Damped Stochastic Wave Equation of Sine–Gordon Type with Sublinear Multiplicative Noise

Abstract

The existence of compact random attractors is proved for a damped stochastic wave equation of Sine–Gordon type with sublinear multiplicative noise under homogeneous Dirichlet boundary condition. To be important, in this note a precise estimate of upper bound of Hausdorff dimension of the random attractors is obtained in lower dimension.     

UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION

In this paper, we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit. As an application,we obtain the convergence of random attractors for non-autonomous stochastic reactiondiffusion equations on unbounded domains, when the density of stochastic noises approaches zero. The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem. A differentiability condition on nonlinearity is omitted, which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity.These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.     

Random attractor for stochastic second-order non-autonomous stochastic lattice equations with dispersive term

In this paper, we consider the asymptotic behaviour of solutions to second-order non-autonomous stochastic lattice equations with dispersive term and additive white noises in the space of infinite sequences. We first transfer the stochastic lattice equations into random lattice equations, and prove the existence and uniqueness of solutions that generate a random dynamical system. Second, we prove the existence of a tempered random absorbing set and a random attractor for the system. Finally, we establish the upper semi-continuity of the random attractors as the coefficient of the white noise term tends to zero.