scholarly journals Upper semicontinuity of the global attractor for the Gierer–Meinhardt model

2006 ◽  
Vol 223 (1) ◽  
pp. 185-207 ◽  
Author(s):  
Yasuhito Miyamoto
Keyword(s):  
Global Attractor ◽  
Upper Semicontinuity

scholarly journals Existence and approximation of attractors for nonlinear coupled lattice wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lianbing She ◽  
Mirelson M. Freitas ◽  
Mauricio S. Vinhote ◽  
Renhai Wang
Keyword(s):  
Asymptotic Behavior ◽  
Global Attractor ◽  
Wave Equations ◽  
Upper Semicontinuity ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Asymptotic Compactness ◽  
Finite Dimensional ◽  
Lattice Wave ◽  
Infinite Lattices

<p style='text-indent:20px;'>This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in <inline-formula><tex-math id="M1">\begin{document}$ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $\end{document}</tex-math></inline-formula>. We then prove that the solution semigroup has a unique global attractor in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula>. We finally prove that this attractor can be approximated in terms of upper semicontinuity of <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula> by a finite-dimensional global attractor of a <inline-formula><tex-math id="M4">\begin{document}$ 2(2n+1) $\end{document}</tex-math></inline-formula>-dimensional truncation system as <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.</p>

UPPER SEMICONTINUITY OF ATTRACTORS FOR THE KLEIN–GORDON–SCHRÖDINGER EQUATION

2005 ◽  
Vol 15 (01) ◽  
pp. 157-168 ◽  
Author(s):  
KENING LU ◽  
BIXIANG WANG
Keyword(s):  
Schrödinger Equation ◽  
Global Attractor ◽  
Schrodinger Equation ◽  
Upper Semicontinuity ◽  
Global Attractors ◽  
Klein Gordon

In this paper, we consider the Klein–Gordon–Schröodinger equation defined on Rn (n ≤ 3) and Ωm = {x ∈ Rn : |x| ≤ m}. Let [Formula: see text] and [Formula: see text] be the global attractors of the equation corresponding to Rn and Ωm, respectively. Then we prove that for any neighborhood U of [Formula: see text], the global attractor [Formula: see text] enters U when m is large enough.

scholarly journals Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation

2005 ◽  
Vol 2005 (6) ◽  
pp. 655-671 ◽  
Author(s):  
Ahmed Y. Abdallah
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Global Attractor ◽  
Fourth Order ◽  
Boussinesq Equation ◽  
Upper Semicontinuity ◽  
Time Variable ◽  
Finite Dimensional ◽  
Lattice Dynamical System ◽  
Second Order In Time

We will study the lattice dynamical system of a nonlinear Boussinesq equation. Our objective is to explore the existence of the global attractor for the solution semiflow of the introduced lattice system and to investigate its upper semicontinuity with respect to a sequence of finite-dimensional approximate systems. As far as we are aware, our result here is the first concerning the lattice dynamical system corresponding to a differential equation of second order in time variable and fourth order in spatial variable with nonlinearity involving the gradients.

scholarly journals DYNAMICS OF A REACTION–DIFFUSION EQUATION WITH A DISCONTINUOUS NONLINEARITY

2006 ◽  
Vol 16 (10) ◽  
pp. 2965-2984 ◽  
Author(s):  
JOSÉ M. ARRIETA ◽  
ANÍBAL RODRÍGUEZ-BERNAL ◽  
JOSÉ VALERO
Keyword(s):  
Nonlinear Dynamics ◽  
Diffusion Equation ◽  
Fixed Points ◽  
Global Attractor ◽  
Nonlinear Term ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Discontinuous Nonlinearity ◽  
Smooth Approximations

We study the nonlinear dynamics of a reaction–diffusion equation where the nonlinearity presents a discontinuity. We prove the upper semicontinuity of solutions and the global attractor with respect to smooth approximations of the nonlinear term. We also give a complete description of the set of fixed points and study their stability. Finally, we analyze the existence of heteroclinic connections between the fixed points, obtaining information on the fine structure of the global attractor.

UPPER SEMICONTINUITY OF GLOBAL ATTRACTORS FOR 2D NAVIER–STOKES EQUATIONS

2012 ◽  
Vol 22 (03) ◽  
pp. 1250046 ◽  
Author(s):  
CAIDI ZHAO ◽  
JINQIAO DUAN
Keyword(s):  
Global Attractor ◽  
Stokes Equations ◽  
Upper Semicontinuity ◽  
Global Attractors ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Simply Connected

In this paper, the authors consider the two-dimensional Navier–Stokes equations defined on Ω = ℝ × (-L, L) and [Formula: see text], where [Formula: see text] is an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that Ωm → Ω as m → ∞. Let [Formula: see text] and [Formula: see text] be the global attractors of the equations corresponding to Ω and Ωm, respectively, we establish that for any neighborhood [Formula: see text] of [Formula: see text], the global attractor [Formula: see text] enters [Formula: see text] if m is large enough.

scholarly journals Existence and upper semicontinuity of global attractors for a $p$-Laplacian inclusion

2014 ◽  
Vol 33 (1) ◽  
pp. 233 ◽  
Author(s):  
Jacson Simsen ◽  
Edson N. Neres Junior
Keyword(s):  
Asymptotic Behavior ◽  
Diffusion Coefficient ◽  
Global Attractor ◽  
Upper Semicontinuity ◽  
Global Attractors ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Diffusion Parameters ◽  
Bounded Smooth ◽  
Convex Subsets

In this work we study the asymptotic behavior of a $p$-Laplacianinclusion of the form $\displaystyle\frac{\partialu_\lambda}{\partial t} - div(D^\lambda|\nablau_\lambda|^{p-2}\nabla u_\lambda) + |u_\lambda|^{p-2}u_\lambda$ $\in F(u_\lambda) + h,$ where $p>2$, $h\in L^2(\Omega),$ with$\Omega\subset\mathbb{R}^n,\; n\geq 1,$ a bounded smooth domain,$D^\lambda \in L^\infty(\Omega)$, $\infty > M\geq D^\lambda(x)\geq \sigma >0$ a.e. in $\Omega$, $\lambda \in [0,\infty)$ and$D^\lambda\rightarrow D^{\lambda_1}$ in $L^\infty(\Omega)$ as$\lambda \to \lambda_1$, $F:\mathcal{D}(F)\subsetL^{2}(\Omega)\rightarrow\mathcal{P}(L^{2}(\Omega))$, given by$F(y(\cdot))=\{\xi(\cdot)\in L^{2}(\Omega):\xi(x)\inf(y(x))\;x\mbox{-a.e. in}\; \Omega\}$ with$f:\mathbb{R}\rightarrow\mathcal{C}_{v}(\mathbb{R})$ Lipschitz($\mathcal{C}_{v}(\mathbb{R})$ is the set of all nonempty,bounded, closed, convex subsets of $\mathbb{R}$) be a multivaluedmap. We prove the existence of a global attractor in $L^2(\Omega)$for each positive finite diffusion coefficient and we show thatthe family of attractors behaves upper semicontinuously onpositive finite diffusion parameters.

ATTRACTORS FOR LATTICE DYNAMICAL SYSTEMS

2001 ◽  
Vol 11 (01) ◽  
pp. 143-153 ◽  
Author(s):  
PETER W. BATES ◽  
KENING LU ◽  
BIXIANG WANG
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Global Attractor ◽  
Upper Semicontinuity ◽  
Global Attractors ◽  
Asymptotic Behavior Of Solutions ◽  
Lattice Differential Equations ◽  
Finite Dimensional ◽  
Lattice Dynamical Systems

We study the asymptotic behavior of solutions for lattice dynamical systems. We first prove asymptotic compactness and then establish the existence of global attractors. The upper semicontinuity of the global attractor is also obtained when the lattice differential equations are approached by finite-dimensional systems.

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