scholarly journals Upper semicontinuity of uniform attractors fornon-autonomous lattice systems under singular perturbations

2021 ◽  
Author(s):  
Lei Na ◽  
Zhou Shengfan
Keyword(s):  
Singular Perturbations ◽  
Upper Semicontinuity ◽  
Lattice Systems ◽  
Uniform Attractors

UNIFORM ATTRACTORS OF NONAUTONOMOUS DISCRETE REACTION–DIFFUSION SYSTEMS IN WEIGHTED SPACES

2008 ◽  
Vol 18 (03) ◽  
pp. 695-716 ◽  
Author(s):  
BIXIANG WANG
Keyword(s):  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Global Attractors ◽  
Weighted Spaces ◽  
Lattice Systems ◽  
Limiting Behavior ◽  
Reaction Diffusion Systems ◽  
Nonautonomous Systems ◽  
Diffusion Systems ◽  
Uniform Attractors

We study the asymptotic behavior of nonautonomous discrete Reaction–Diffusion systems defined on multidimensional infinite lattices. We show that the nonautonomous systems possess uniform attractors which attract all solutions uniformly with respect to the translations of external terms when time goes to infinity. These attractors are compact subsets of weighted spaces, and contain all bounded solutions of the system. The upper semicontinuity of the uniform attractors is established when an infinite-dimensional reaction–diffusion system is approached by a family of finite-dimensional systems. We also examine the limiting behavior of lattice systems with almost periodic, rapidly oscillating external terms in weighted spaces. In this case, it is proved that the uniform global attractors of nonautonomous systems converge to the global attractor of an averaged autonomous system.

scholarly journals Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Na Lei ◽  
Shengfan Zhou
Keyword(s):  
Singular Perturbations ◽  
Upper Semicontinuity ◽  
Second Order ◽  
Second Derivative ◽  
Compact Sets ◽  
Pullback Attractors ◽  
Lattice Systems ◽  
First Order ◽  
Derivative Term

<p style='text-indent:20px;'>Consider the second order nonautonomous lattice systemswith singular perturbations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \epsilon \ddot{u}_{m}+\dot{u}_{m}+(Au)_{m}+\lambda_{m}u_{m}+f_{m}(u_{j}|j\in I_{mq}) = g_{m}(t),\; \; m\in \mathbb{Z}^{k},\; \; \epsilon&gt;0 \tag{*} \label{0} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the first order nonautonomous lattice systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \dot{u}_{m}+(Au)_{m}+\lambda _{m}u_{m}+f_{m}(u_{j}|j∈I_{mq}) = g_{m}(t),\; \; m\in \mathbb{Z}^{k}. \tag{**} \label{00} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Under certain conditions, there are pullback attractors <inline-formula><tex-math id="M1">\begin{document}$ \{\mathcal{A}_{\epsilon }(t)\subset \ell ^{2}\times \ell ^{2}\}_{t\in \mathbb{R}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \{\mathcal{A}(t)\subset \ell ^{2}\}_{t\in \mathbb{R}} $\end{document}</tex-math></inline-formula> for systems (*)and (**), respectively. In this paper, we mainly consider the uppersemicontinuity of attractors <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{A}_{\epsilon }(t)\subset \ell^{2}\times \ell ^{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula>, with respect to the coefficient <inline-formula><tex-math id="M5">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> of second derivative term under Hausdorff semidistance. First, we studythe relationship between <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{A}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M8">\begin{document}$ \epsilon \rightarrow 0^{+} $\end{document}</tex-math></inline-formula>. We construct a family of compact sets <inline-formula><tex-math id="M9">\begin{document}$ \mathcal{A}_{0}(t)\subset \ell ^{2}\times \ell ^{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M11">\begin{document}$ \mathcal{A}(t) $\end{document}</tex-math></inline-formula> is naturally embedded into <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{A}_{0}(t) $\end{document}</tex-math></inline-formula> as the firstcomponent, and prove that <inline-formula><tex-math id="M13">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> can enter anyneighborhood of <inline-formula><tex-math id="M14">\begin{document}$ \mathcal{A}_{0}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M15">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is small enough. Thenfor <inline-formula><tex-math id="M16">\begin{document}$ \epsilon _{0}&gt;0 $\end{document}</tex-math></inline-formula>, we prove that <inline-formula><tex-math id="M17">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> can enterany neighborhood of <inline-formula><tex-math id="M18">\begin{document}$ \mathcal{A}_{\epsilon _{0}}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M19">\begin{document}$ \epsilon\rightarrow \epsilon _{0} $\end{document}</tex-math></inline-formula>. Finally, we consider the existence andexponentially attraction of the singleton pullback attractors of systems (*)-(**).</p>

Upper semicontinuity of attractors for lattice systems under singular perturbations

2010 ◽  
Vol 72 (5) ◽  
pp. 2149-2158 ◽  
Author(s):  
Caidi Zhao ◽  
Shengfan Zhou
Keyword(s):  
Singular Perturbations ◽  
Upper Semicontinuity ◽  
Lattice Systems

scholarly journals Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems

2008 ◽  
Vol 21 (4) ◽  
pp. 1259-1277 ◽  
Author(s):  
Shengfan Zhou ◽  
◽  
Caidi Zhao ◽  
Yejuan Wang ◽  
◽  
...  
Keyword(s):  
Upper Semicontinuity ◽  
Lattice Systems ◽  
Finite Dimensionality

Wong–Zakai approximations of second-order stochastic lattice systems driven by additive white noise

2021 ◽  
pp. 2150050
Author(s):  
Yiju Chen ◽  
Chunxiao Guo ◽  
Xiaohu Wang
Keyword(s):  
White Noise ◽  
Additive Noise ◽  
Upper Semicontinuity ◽  
Second Order ◽  
Pullback Attractors ◽  
Lattice Systems ◽  
Additive White Noise ◽  
Random Attractors ◽  
Approximate System

In this paper, we study the Wong–Zakai approximations of a class of second-order stochastic lattice systems with additive noise. We first prove the existence of tempered pullback attractors for lattice systems driven by an approximation of the white noise. Then, we establish the upper semicontinuity of random attractors for the approximate system as the size of approximation approaches zero.

scholarly journals Existence and Upper Semicontinuity of Attractors for Nonautonomous Stochastic Sine-Gordon Lattice Systems with Random Coupled Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Zhaojuan Wang ◽  
Shengfan Zhou
Keyword(s):  
White Noise ◽  
Weighted Space ◽  
Upper Semicontinuity ◽  
Lattice Systems ◽  
Random Attractors ◽  
Multiplicative White Noise

We study nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise. We first consider the existence of random attractors in a weighted space for this system and then establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

scholarly journals Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yiju Chen ◽  
Xiaohu Wang
Keyword(s):  
Asymptotic Behavior ◽  
Multiplicative Noise ◽  
Upper Semicontinuity ◽  
Random Attractor ◽  
Discrete Laplacian ◽  
Lattice Systems ◽  
Random Attractors ◽  
Infinite Range Interactions ◽  
Infinite Range

<p style='text-indent:20px;'>In this paper, we study the asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise. The considered systems are driven by the fractional discrete Laplacian, which features the infinite-range interactions. We first prove the existence of pullback random attractor in <inline-formula><tex-math id="M1">\begin{document}$ \ell^2 $\end{document}</tex-math></inline-formula> for stochastic lattice systems. The upper semicontinuity of random attractors is also established when the intensity of noise approaches zero.</p>

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