Non-autonomous Lattice Systems Research Articles

Random Uniform Attractors for First Order Stochastic Non-Autonomous Lattice Systems

Random Uniform Attractors for First Order Stochastic Non-Autonomous Lattice Systems

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

<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>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}>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 uniform attractors fornon-autonomous lattice systems under singular perturbations

In this paper, we study the upper semicontinuity ofthe uniform attractors for the second order nonautonomous lattice systemsdriven almost periodic force with respect tothe coefficient of the second order derivative term under Hausdorffsemidistance. First, we present the existence of uniform attractors for thesecond order nonautonomous lattice systems and the corresponding first order nonautonomous lattice systems under certain 条件, respectively. Then we consider the relationship between these two uniform attractors when the coefficient of the second derivation term tendsto zero. The method is to construct a new set in the phase space of thesecond order lattice systems such that the uniform attractorof the first order lattice system is naturally embedded intothis set as the first component, and prove that the uniform attractors forthe second order nonautonomous lattice system can enter any neighborhood of thisset when the coefficient of the second derivation term is small enough. Finally,we discuss the upper semicontinuity of the uniform attractors of the secondorder nonautonomous lattice systems at any positive coefficient of the secondderivative term.

On the forward dynamical behaviour of nonautonomous lattice dynamical systems

In this article, we study the forward dynamical behaviour of nonautonomous lattice systems. We first construct a family of sets in the arbitrary small neighbourhood of a global attractor of the skew-product flow generated by a general nonautonomous lattice system, which is forward invariant and uniformly forward attracts any bounded subset of the phase space. Moreover, under some suitable conditions, we further construct a family of sets such that it uniformly forward exponentially attracts bounded subsets of the phase space. As an application, we study the discrete Gray–Scott model in detail and illustrate how to apply our abstract results to some concrete lattice system.

Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients

We study longtime behavior for the non-autonomous lattice model with multiplicative white noise and a random coefficient in the discrete Laplace operator. We first show existence of a bi-spatial attractor when the initial space is the weighted square summation space and the terminal space is the weighted p-times summation space for p > 2. We then show backward compactness of the attractor in both initial and terminal spaces if the time-indexed forces are backward-tempered and backward-null. Finally, by proving identity of the attractors on the different universes of tempered or backward tempered sets, we show measurability of the attractor in the initial space and in the terminal space, respectively.

Second order non-autonomous lattice systems and their uniform attractors

The existence of the uniform global attractor for a second order non-autonomous lattice dynamical system (LDS) with almost periodic symbols has been carefully studied. Considering the nonlinear operators $ \left( f_{1i}\left( \overset{.}{u}_{j}\mid j\in I_{iq_{1}}\right) \right) _{i\in \mathbb{Z} ^{n}} $ and $ \left( f_{2i}\left( u_{j}\mid j\in I_{iq_{2}}\right) \right) _{i\in \mathbb{Z} ^{n}} $ of this LDS, up to our knowledge it is the first time to investigate the existence of uniform global attractors for such second order LDSs. In fact there are some previous studies for first order autonomous and non-autonomous LDSs with similar nonlinear parts, cf. [3, 24]. Moreover, the LDS under consideration covers a wide range of second order LDSs. In fact, for specific choices of the nonlinear functions $ f_{1i} $ and $ f_{2i} $ we get the autonomous and non-autonomous second order systems given by [1, 25, 26].

Non-autonomous lattice systems with switching effects and delayed recovery

The long term behavior of a type of non-autonomous lattice dynamical systems is investigated, where these have a diffusive nearest neighborhood interaction and discontinuous reaction terms with recoverable delays. This problem is of both biological and mathematical interests, due to its application in systems of excitable cells as well as general biological systems involving delayed recovery. The problem is formulated as an evolution inclusion with delays and the existence of weak and strong solutions is established. It is then shown that the solutions generate a set-valued non-autonomous dynamical system and that this non-autonomous dynamical system possesses a non-autonomous global pullback attractor.

Attractors for second order nonautonomous lattice system with dispersive term

In this paper, we prove the existence of pullback attractor, pullback exponential attractor and uniform attractor for second order non-autonomous lattice system with dispersive term and time-dependent forces. Then we prove the existence of uniform exponential attractor for the system driven by quasi-periodic external forces.

Global attractors for the discrete Klein–Gordon–Schrödinger type equations

The purpose of this work is to investigate the asymptotic behaviours of solutions for the discrete Klein–Gordon–Schrödinger type equations in one-dimensional lattice. We first establish the global existence and uniqueness of solutions for the corresponding Cauchy problem. According to the solution's estimate, it is shown that the semi-group generated by the solution is continuous and possesses an absorbing set. Using truncation technique, we show that there exists a global attractor for the semi-group. Finally, we extend the criteria of Zhou et al. [S. Zhou, C. Zhao, and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst. A 21 (2008), pp. 1259–1277.] for finite fractal dimension of a family of compact subsets in a Hilbert space to obtain an upper bound of fractal dimension for the global attractor.

Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

We first present some sufficient conditions for the existence of a pullback exponential attractor for continuous process on the product space of the weighted spaces of infinite sequences. Then we prove the existence and continuity of a pullback exponential attractor for second order lattice system with time-dependent coupled coefficients in the weighted space of infinite sequences. Moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.

Pullback Exponential Attractors for Non-autonomous Lattice Systems

We first present some sufficient conditions for the existence and the construction of a pullback exponential attractor for the continuous process (non-autonomous dynamical system) on Banach spaces and weighted spaces of infinite sequences. Then we apply our results to study the existence of pullback exponential attractors for first order non-autonomous differential equations and partly dissipative differential equations on infinite lattices with time-dependent coupled coefficients and time-dependent external terms in weighted spaces.

Uniform Attractor for the Partly Dissipative Nonautonomous Lattice Systems

The existence of uniform attractor in is proved for the partly dissipative nonautonomous lattice systems with a new class of external terms belonging to , which are locally asymptotic smallness and translation bounded but not translation compact in . It is also showed that the family of processes corresponding to nonautonomous lattice systems with external terms belonging to weak topological space possesses uniform attractor, which is identified with the original one. The upper semicontinuity of uniform attractor is also studied.

Compact kernel sections for lattice systems with delays

In this paper, we consider the existence, upper semi-continuity and singular limiting behavior of compact kernel sections for the process associated to non-autonomous lattice systems with delays.

Kernel sections for processes and nonautonomous lattice systems

In this paper, we first establish a set of sufficient and necessaryconditions for the existence of globally attractive kernel sectionsfor processes defined on a general Banach space and a weighted spaceℓ$_\rho ^p$ of infinite sequences ($p\geq 1)$, respectively.Then we obtain an upper bound of the Kolmogorov$\varepsilon$-entropy of kernel sections for processes on theHilbert space ℓ$_\rho ^2 $. As applications, we investigatecompact kernel sections for first order, partly dissipative, andsecond order nonautonomous lattice systems on weighted spacescontaining bounded sequences.

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

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.

Asymptotic behavior of non-autonomous lattice systems

The asymptotic behavior of non-autonomous infinite-dimensional lattice systems is studied. It is shown that the non-autonomous lattice reaction–diffusion system has a compact uniform attractor. The uniform asymptotic compactness of the system is established by showing that the tails of the solutions are uniformly small when time goes to infinity. The upper semicontinuity of uniform attractors is also obtained when the infinite-dimensional reaction–diffusion system is approached by a family of finite-dimensional systems.