DCDS-B
Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises
Min Zhao Shengfan Zhou
Discrete & Continuous Dynamical Systems - B 2017, 22(4): 1683-1717 doi: 10.3934/dcdsb.2017081

In this paper, we study the long-term dynamical behavior of stochastic Boisso nade systems with time-dependent deterministic forces, additive white noise and multiplicative white noise. We first prove the existence of random attrac tor for the considered systems. And then we establish the upper semi-continui ty of random attractors for the systems as the coefficient of quadratic term tends to zero and intensities of the noises approach zero, respectively. At last, we obtain an upper bound of fractal dimension of the random attractors for both systems without quadratic term.

keywords: Random attractor stochastic Boissonade system additive white noise multiplicative white noise upper semi-continuity fractal dimension
DCDS-B
Kernel sections for processes and nonautonomous lattice systems
Xiao-Qiang Zhao Shengfan Zhou
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 763-785 doi: 10.3934/dcdsb.2008.9.763
In this paper, we first establish a set of sufficient and necessary conditions for the existence of globally attractive kernel sections for processes defined on a general Banach space and a weighted space ℓ$_\rho ^p$ of infinite sequences ($p\geq 1)$, respectively. Then we obtain an upper bound of the Kolmogorov $\varepsilon$-entropy of kernel sections for processes on the Hilbert space ℓ$_\rho ^2 $. As applications, we investigate compact kernel sections for first order, partly dissipative, and second order nonautonomous lattice systems on weighted spaces containing bounded sequences.
keywords: Kernel sections lattice systems pullback asymptotic null. pullback $\omega $-limit compactness Kolmogorov's $\varepsilon $-entropy
DCDS
Random attractor for stochastic non-autonomous damped wave equation with critical exponent
Zhaojuan Wang Shengfan Zhou
Discrete & Continuous Dynamical Systems - A 2017, 37(1): 545-573 doi: 10.3934/dcds.2017022

In this paper, we prove the existence of random attractor and obtainan upper bound of fractal dimension of random attractor forstochastic non-autonomous damped wave equation with criticalexponent and additive white noise. We first prove the existence of arandom attractor by carefully splitting the positivity of the linearoperator in the corresponding random evolution equation of the firstorder in time and by carefully decomposing the solutions of systemthrough two different modes, and we show the boundedness of randomattractor in a higher regular space by a recurrence method. Then weestablish a criterion to bound the fractal dimension of a randominvariant set for a cocycle and applied these conditions to get anupper bound of fractal dimension of the random attractor ofconsidered system.

keywords: Stochastic damped wave equation random attractor fractal dimension critical exponent
DCDS
Compact uniform attractors for dissipative lattice dynamical systems with delays
Caidi Zhao Shengfan Zhou
Discrete & Continuous Dynamical Systems - A 2008, 21(2): 643-663 doi: 10.3934/dcds.2008.21.643
In this paper, we consider the long time behavior of solutions for dissipative lattice dynamical systems with delays. We first prove a sufficient and necessary condition for the existence of a compact uniform attractor for the family of processes corresponding to the lattice dynamical systems with delays. Then we apply this result to prove the existence of a compact uniform attractor for the process associated to the retarded lattice Zakharov equations. As a consequence, some results for the non-delay lattice dynamical systems are deduced as particular cases.
keywords: Lattice dynamical systems; Uniform attractor; Delay systems; Kolmogorov $\varepsilon$-entropy.
DCDS
Kernel sections for damped non-autonomous wave equations with critical exponent
Shengfan Zhou Linshan Wang
Discrete & Continuous Dynamical Systems - A 2003, 9(2): 399-412 doi: 10.3934/dcds.2003.9.399
We prove the existence of kernel sections for the process generated by a damped non-autonomous wave equation when there is nonlinear damping and the nonlinearity has a critically growing exponent. We show uniform boundedness of the Hausdorff dimension of the kernel sections. Finally, we point out that in the case of autonomous systems with linear damping, the obtained upper bound of the Hausdorff dimension decreases as the damping grows for suitable large damping.
keywords: kernel section Hausdorff dimension. process Wave equation
DCDS
Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise
Zhaojuan Wang Shengfan Zhou
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4767-4817 doi: 10.3934/dcds.2018210

In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.

keywords: Stochastic damped wave equation random attractor random exponential attractor multiplicative white noise upper semicontinuity fractal dimension regularity
DCDS
Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise
Zhaojuan Wang Shengfan Zhou
Discrete & Continuous Dynamical Systems - A 2017, 37(5): 2787-2812 doi: 10.3934/dcds.2017120

In this paper we study the asymptotic behavior of solutions of the non-autonomous stochastic strongly damped wave equation driven by multiplicative noise defined on unbounded domains. We first introduce a continuous cocycle for the equation. Then we consider the existence of a tempered pullback random attractor for the cocycle. Finally we establish the upper semicontinuity of random attractors as the coefficient of the white noise term tends to zero.

keywords: Stochastic strongly damped wave equation unbounded domains random attractor upper semicontinuity
CPAA
Compact uniform attractors for dissipative non-autonomous lattice dynamical systems
Xinyuan Liao Caidi Zhao Shengfan Zhou
Communications on Pure & Applied Analysis 2007, 6(4): 1087-1111 doi: 10.3934/cpaa.2007.6.1087
This paper discusses the long time behavior of solutions for dissipative non-autonomous lattice dynamical systems. We first prove some sufficient and necessary conditions for the existence of a compact uniform attractor for the family of processes defined on a Hilbert space of infinite sequences, and then give an upper bound of the Kolmogorov $\varepsilon$-entropy for the uniform attractor. As an application, we consider the dissipative non-autonomous lattice Zakharov equations.
keywords: lattice Zakharov equations Kolmogorov $\varepsilon$-entropy Lattice dynamical systems non-autonomous. uniform attractor
DCDS
Pullback attractors of nonautonomous dynamical systems
Yejuan Wang Chengkui Zhong Shengfan Zhou
Discrete & Continuous Dynamical Systems - A 2006, 16(3): 587-614 doi: 10.3934/dcds.2006.16.587
We present the necessary and sufficient conditions and a new method to study the existence of pullback attractors of nonautonomous infinite dimensional dynamical systems. For illustrating our method, we apply it to nonautonomous 2D Navier-Stokes systems. We also show that the parametrically inflated pullback attractors and uniform attractors are robust with respect to the perturbations of both cocycle mappings and driving systems. As an example, we consider the nonautonomous 2D Navier-Stokes system with rapidly oscillating external force.
keywords: 2D Navier-Stokes system. pullback attractor nonautonomous dynamical system
DCDS
Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems
Shengfan Zhou Caidi Zhao Yejuan Wang
Discrete & Continuous Dynamical Systems - A 2008, 21(4): 1259-1277 doi: 10.3934/dcds.2008.21.1259
In this paper, we first establish a criteria for finite fractal dimensionality of a family of compact subsets of a Hilbert space, and apply it to obtain an upper bound of fractal dimension of compact kernel sections to first order non-autonomous lattice systems. Then we consider the upper semicontinuity of kernel sections of general first order non-autonomous lattice systems and give an application.
keywords: Upper semicontinuity. Kernel section Fractal dimension Non-autonomous lattice system

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