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.     

一类非自治发展方程的一致吸引子

本文研究一类非自治发展方程的渐近行为,运用算子分解及分析技巧得到了系统解的渐近正则性,由此证明一致吸引子的存在性、正则性及其结构.其中非线性项满足临界指数增长,时间依赖的外力项仅假设是平移有界而不是平移紧的.     

带有乘积白噪音的非自治随机波方程的随机吸引子

研究了带有乘积白噪音的非自治随机波方程.首先证明解在一个有界球外的一致小性,然后对解在有界的区域内进行分解,得到解的渐近紧性,最后得到了带有乘积白噪音的非自治随机波方程的随机吸引子的存在性.     

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.     

色噪声驱动的非自治非经典扩散方程的拉回随机吸引子

本文研究无界区域上带色噪声的非自治非经典扩散方程解的渐近性.为了克服Sobolev嵌入在无界区域上的非紧性,利用Ball能量方程的思想,本文证明与问题相关的多值随机动力系统在H¹(Rⁿ)中拉回随机吸引子的存在性.     

随机广义Ginzburg-Landau方程的吸引子

可以按轨道得到带白噪声的随机广义Ginzburg-Landau方程的唯一解并且能够验证该解可以产生随机系统, 从而证明了该随机系统在H¹₀中存在整体随机吸引子.     

非自治的Schroedinger方程的吸引子

研究2维的非自治非线性Schroedinger方程长时间的动力学行为。证明了一致吸引子的存在性,并给出了该一致吸引子Hausdorff维数的上界。     

随机耗散Camassa-Holm方程的吸引子

可以接轨道得到带白噪声的随机耗散Camassa-Holm方程的唯—解并且可以检验该解产生随机动力系统,从而证明了该随机动力系统在H₀²中存在紧的随机吸引子.     

一类非自治时滞抛物方程的一致吸引子

本文研究了一类具有平移有界非自治外力的时滞抛物方程的动力学性态,得到了其整体解的存在和唯一性.利用时间符号理论,并通过构造乘积空间中的紧吸收集,在时间符号空间非紧的情形下得到了一致吸引子的存在性.     

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.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Ginzburg–Landau Equations

This paper considers the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg–Landau equations driven by additive noise with α∈(0, 1). First, we give some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Second, we derive uniform estimates of solutions and establish the existence and uniqueness of tempered pullback random attractors for the equation in H. At last, we prove the finiteness of fractal dimension of random attractors.     

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.     

Random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations on unbounded domains

This paper deals with the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with $\alpha\in(0,1)$. We prove the existence and uniqueness of tempered pullback random attractors for the equations in $L^{2}(\mathbf{R}^{3})$. In addition, we also obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. The main difficulty here is the noncompactness of Sobolev embeddings on unbounded domains. To solve this, we establish the pullback asymptotic compactness of solutions in $L^{2}(\mathbf{R}^{3})$ by the tail-estimates of solutions.     

Random Attractors for the Stochastic Kuramoto-Sivashinsky Equation

Abstract

Random systems may be more reasonable by incorporating influence of noise into deterministic systems. The notion of a random attractor is one of the very basic concepts of the theory of random dynamical systems. In this article, we consider the well-known Kuramoto–Sivashinsky equation with stochastic perturbation. Our aim is to attempt to obtain a so-called pull-back random attractor for stochastic Kuramoto–Sivashinsky equation. In particular, the Hausdorff dimension of a random attractor is finite. For simplicity, we always restrict ourselves to odd initial conditions, but the result for all initial conditions is also true.     

关于局部随机吸引子若干性质的注记(英文)

In this note, we study some properties of local random pull-back attractors on compact metric spaces. We obtain some relations between attractors and their fundamental neighborhoods and basins of attraction. We also obtain some properties of omega-limit sets, as well as connectedness of random attractors. A simple deterministic example is given to illustrate some confusing problems.     

The Global Random Attractor for a Class of Stochastic Porous Media Equations

We prove new L ²-estimates and regularity results for generalized porous media equations “shifted by” a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of “ζ-monotonicity” for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.     

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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 derPol方程随机吸引子的存在性和随机分岔.     

Pullback Attractors of Nonautonomous and Stochastic Multivalued Dynamical Systems

In this paper we study the existence of pullback global attractors for multivalued processes generated by differential inclusions. First, we define multivalued dynamical processes, prove abstract results on the existence of -limit sets and global attractors, and study their topological properties (compactness, connectedness). Further, we apply the abstract results to nonautonomous differential inclusions of the reaction–diffusion type in which the forcing term can grow polynomially in time, and to stochastic differential inclusions as well.