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.     

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.     

渐近上半紧的多值随机半流的随机吸引子

该文研究多值随机半流的随机吸引子的存在性．首先证明在拉回渐近上半紧及吸收的条件下,关于极限集的一个抽象结果,然后证明了随机的吸引子的存在性．     

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.     

连续随机变换的Lyapunov指数

本文对连续随机变换引入了Lyapunov指数的概念．对一类由扰动相应确定系统而产生的随机排斥子和随机双曲集而言,该概念和经典的概念是一致的．     

非自治薛定谔格点方程的拉回和一致指数吸引子

主要考虑非自治薛定谔格点系统的拉回指数吸引子和一致指数吸引子的存在性以及它们的分形维数.首先,证明具时变耦合系数的薛定谔格点系统在依时间外力作用下的拉回指数吸引子的存在性;然后,证明拟周期外力驱动下的非自治薛定谔格点系统的一致指数吸引子的存在性。     

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.     

非自治Boissonade系统的拉回和一致指数吸引子

本文首先改进已有的非自治动力系统的拉回和一致指数吸引子的存在性条件;然后证明受依赖时间外力扰动的Boissonade系统的拉回指数吸引子的存在性、连续性和分形维数有限性;最后证明受拟周期外力扰动的Boissonade系统的一致指数吸引子存在性及其分形维数有限性.     

Random attractor and upper semi-continuity for Zakharov lattice system with multiplicative white noise

In this paper, we study the long-term asymptotic behaviour of solutions to the stochastic Zakharov lattice equation with multiplicative white noise. We first transfer the stochastic lattice equation into a random lattice equation and prove the existence and uniqueness of solutions which generate a random dynamical system. Then we consider the existence of a tempered random bounded absorbing set and a random attractor for the system. Finally we establish the upper semi-continuity of random attractor to the global attractor of the limiting system as the coefficients of the white noise terms tend to zero.     

Random attractor of stochastic complex Ginzburg–Landau equation with multiplicative noise on unbounded domain

The existence of a compact random attractor for the stochastic complex Ginzburg–Landau equation with multiplicative noise has been investigated on unbounded domain. The solutions are considered in suitable spaces with weights. According to crucial properties of Ornstein–Uhlenbeck process, using the tail-estimates method, the key uniform a priori estimates for the tail of solutions have been obtained, which give the asymptotic compactness of random attractors. Then the existence of a compact random attractor for the corresponding dynamical system is proved in suitable spaces with weights.     

带加法白噪声的随机Boussinesq方程组的解的渐近行为

考虑速度和温度同时在加法白噪声扰动下的随机Boussinesq方程组的解的渐近特征.可以接轨道得到该随机方程组的唯一解,并可以验证该解生成随机动力系统,进而证明了该随机动力系统存在随机吸引子.     

Random attractor of stochastic Zakharov lattice system

In this paper, we consider a lattice system of stochastic Zakharov equation with white noise. We first show that the solutions of the system determine a continuous random dynamical system with random absorbing set. And then we prove the random  asymptotic compact on the random absorbing set. Finally, we obtain the existence of a random attractor for the system.     

Uniform exponential attractors for second order non-autonomous lattice dynamical systems

In this paper, the existence of a uniform exponential attractor for a second order non-autonomous lattice dynamical system with quasiperiodic symbols acting on a closed bounded set is considered. Firstly, the existence and uniqueness of solutions for the considered systems which generate a family of continuous processes is shown, and the existence of a uniform bounded absorbing sets for the processes is proved. Secondly, a semigroup defined on a extended space is introduced, and the Lipschitz continuity, α-contraction and squeezing property of this semigroup are proved. Finally, the existence of a uniform exponential attractor for the family of processes associated with the studied lattice dynamical systems is obtained.     

Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains

In this paper we study the asymptotic dynamics for stochastic reaction-diffusion equation with multiplicative noise defined on unbounded domains. We investigate the existence of a random attractor for the random dynamical system associated with the equation. The asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far-field values of solutions and a cut-off technique.     

Random attractors of stochastic partial differential equations: A smooth approximation approach

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.     

Pullback exponential attractors for nonautonomous equations Part II: Applications to reaction-diffusion systems

The existence of a pullback exponential attractor being a family of compact and positively invariant sets with a uniform bound on their fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the evolution process is proved for the nonautonomous logistic equation and a system of reaction-diffusion equations with time-dependent external forces including the case of the FitzHugh-Nagumo system.     

Asymptotic autonomous attractors for a stochastic lattice model with random viscosity

We study forward asymptotic autonomy of a pullback random attractor for a non-autonomous random lattice system and establish the criteria in terms of convergence, recurrence, forward-pullback absorption and asymptotic smallness of the discrete random dynamical system. By applying the abstract result to both non-autonomous and autonomous stochastic lattice equations with random viscosity, we show the existence of both pullback and global random attractors such that the time-component of the pullback attractor semi-converges to the global attractor as the time-parameter tends to infinity.