DCDS-B
Kernel sections for processes and nonautonomous lattice systems
Xiao-Qiang Zhao Shengfan Zhou
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

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
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
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
Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise
Zhaojuan Wang Shengfan Zhou

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
DCDS-B
Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises
Min Zhao Shengfan Zhou

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
CPAA
Compact kernel sections for dissipative non-autonomous Zakharov equation on infinite lattices
Shengfan Zhou Jinwu Huang Xiaoying Han
In this paper, we first prove the existence of compact kernel sections for dissipative non-autonomous Zakharov lattice dynamical system, then we obtain an upper bound of fractal dimension of the compact kernel sections, and finally we establish an upper semicontinuity of the compact kernel sections.
keywords: non-autonomous fractal dimension Zakharov lattice system. upper semicontinuity Compact kernel sections
DCDS
The existence of uniform attractors for 3D Brinkman-Forchheimer equations
Yuncheng You Caidi Zhao Shengfan Zhou
The longtime dynamics of the three dimensional (3D) Brinkman-Forchheimer equations with time-dependent forcing term is investigated. It is proved that there exists a uniform attractor for this nonautonomous 3D Brinkman-Forchheimer equations in the space $\mathbb{H}^1(\Omega)$. When the Darcy coefficient $\alpha$ is properly large and $L^2_b$-norm of the forcing term is properly small, it is shown that there exists a unique bounded and asymptotically stable solution with interesting corollaries.
keywords: uniform attractor. Brinkman-Forchheimer equation asymptotic dynamics
CPAA
Compact uniform attractors for dissipative non-autonomous lattice dynamical systems
Xinyuan Liao Caidi Zhao Shengfan Zhou
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
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

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