scholarly journals Global Attractors for the Higher-Order Evolution Equation

2020 ◽  
Vol 5 (1) ◽  
pp. 195-210
Author(s):  
Erhan Pişkin ◽  
Hazal Yüksekkaya
Keyword(s):  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Global Solution ◽  
Global Attractor ◽  
Type Equation ◽  
Global Attractors ◽  
Higher Order ◽  
Evolution Type

AbstractIn this paper, we obtain the existence of a global attractor for the higher-order evolution type equation. Moreover, we discuss the asymptotic behavior of global solution.

scholarly journals The Global and Exponential Attractors for the Higher-order Kirchhoff-type Equation with Strong Linear Damping

2017 ◽  
Vol 9 (4) ◽  
pp. 145 ◽  
Author(s):  
Guoguang Lin ◽  
Yunlong Gao
Keyword(s):  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Attractors ◽  
Higher Order ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Kirchhoff Type ◽  
Uniqueness Of The Solution ◽  
Initial Boundary Value

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: ${u_{tt}} + {( - \Delta )^m}{u_t} + {\left( {\alpha + \beta\left\| {{\nabla ^m}u} \right\|^2} \right)^{q}}{( - \Delta )^m}u + g(u) = f(x)$. At first, we do priori estimation for the equations to obtain two lemmas and prove the existence and uniqueness of the solution by the lemmas and the Galerkin method. Then, we obtain to the existence of the global attractor in $H_0^m(\Omega ) \times {L^2}(\Omega )$ according to some of the attractor theorem. In this case, we consider that the estimation of the upper bounds of Hausdorff  for the global attractors are obtained. At last, we also establish the existence of a fractal exponential attractor with the non-supercritical and critical cases.

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.

scholarly journals The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation*

2016 ◽  
Vol 12 (9) ◽  
pp. 6608-6621
Author(s):  
Ling Chen ◽  
Wei Wang ◽  
Guoguang Lin
Keyword(s):  
Existence And Uniqueness ◽  
Finite Dimension ◽  
Fractal Dimensions ◽  
Type Equation ◽  
Global Attractors ◽  
Higher Order ◽  
Well Posedness ◽  
Priori Estimate ◽  
Uniqueness Of The Solution ◽  
Hausdorff And Fractal Dimensions

We investigate the global well-posedness and the longtime dynamics of solutions for the higher-order Kirchhoff-typeequation with nonlinear strongly dissipation:2( ) ( )m mt t tu    u    D u  ( ) ( ) ( )m  u  g u  f x . Under of the properassume, the main results are that existence and uniqueness of the solution is proved by using priori estimate and Galerkinmethod, the existence of the global attractor with finite-dimension, and estimation Hausdorff and fractal dimensions of theglobal attractor.

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