Uniform attractors for the closed process and applications to the reaction-diffusion equation with dynamical boundary condition
type: Item
instance of: scholarly article
title: Uniform attractors for the closed process and applications to the reaction-diffusion equation with dynamical boundary condition
zbMATH Open document ID: 1173.37061
author: Yang Lu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-07-10T00:00:00Z
review text: The author studies parametrized families of processes \[ {\mathcal U}_\sigma(t,\tau),\quad t\geq \tau,\;\sigma\in\Sigma\tag{1} \] which are associated with nonautonomous semilinear parabolic equations. Here, \({\mathcal U}_\sigma(t,\tau)\) is a bounded operator, acting on a Banach space \(X\) such that: \[ \begin{aligned} & {\mathcal U}_\sigma(t,s){\mathcal U}_\sigma(s,\tau)={\mathcal U}_\sigma(t,\tau),\quad t\geq s\geq \tau,\;\sigma\in\Sigma\tag{a}\\ & U_\sigma(s,s)=I,\quad s\in \mathbb R,\;\sigma\in\Sigma.\tag{b}\end{aligned} \] The symbol space \(\Sigma\) is a metric space with metric \(d(\;)\), on which a translation semigroup \(T(\;)\) acts according to: \[ {\mathcal U}_{T(s)\sigma}(t,\tau)={\mathcal U}_\sigma(t+s,\tau+s),\quad t\geq\tau,\;\sigma\in\Sigma,\;s\geq 0.\tag{2} \] The author recalls a series of definitions related to parametrized processes. Definition 2.3 eg. introduces the notion of an attractor of a parametrized process via (1), uniform with respect to \(\sigma\in\Sigma\). A new concept is introduced by Definition 3.1 in which the notion of closed parametrized family of processes is defined. The notion of closed translation semigroup \(T(s)\), \(s\geq 0\) is then defined by Definition 3.2. The author then proceeds to the proof of some theorems concerning this abstract setting. Theorem 3.5 eg. asserts the existence of an attractor for the parametrized process (1), uniform with respect to \(\sigma\in\Sigma\), provided that \(\Sigma\) is compact, the process (1) and the semigroup \(T(s)\), \(s\geq 0\) closed, and some assumptions of technical nature satisfied. In sect. 5 the abstract theory is applied to a non-autonomous reaction-diffusion equation with dynamic boundary conditions, ie.: \[ \begin{alignedat}{2} & u_t-\Delta u+f(u)=h(t),&\quad & \text{on }[\tau,T]\times \Omega\tag{3}\\ & u_t+\frac{\partial u}{\partial n}+g(u)=0,&\quad & \text{on }(\tau,T)\times \partial\Omega,\\ &u(x,\tau)=u_\tau(x) &\quad & \text{on }\Omega\subseteq R^n.\end{alignedat…
reviewed by: Bruno Scarpellini
Mathematics Subject Classification ID: 37L05
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 5577466
zbMATH Keywords: uniform attractor
zbMATH Keywords: closed process
zbMATH Keywords: skew productively closed
zbMATH Keywords: kernel section
zbMATH Keywords: reaction-diffusion equation
zbMATH Keywords: dynamical boundary condition
MaRDI profile type: Publication
full work available at URL: j.na.2009.02.083
OpenAlex ID: W2005786062
cites work: Attractors for quasilinear second-order parabolic equations of the general form
cites work: Infinite-dimensional dynamical systems in mechanics and physics.
cites work: Q4003101
cites work: Introduction (to ``From finite to infinite dimensional dynamical systems'')
cites work: Dynamics of evolutionary equations
cites work: Q4039920
cites work: Q3997729
cites work: Attractors for equations of mathematical physics
cites work: Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces
cites work: Attractors for noncompact nonautonomous systems via energy equations.
cites work: The attractor for a nonlinear hyperbolic equation in the unbounded domain
cites work: The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations
cites work: Uniform Attractors for Nonautonomous Wave Equations with Nonlinear Damping
cites work: Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data
cites work: Quasilinear parabolic systems with dynamical boundary conditions
cites work: Attractors for parabolic equations with dynamic boundary conditions
cites work: Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions
cites work: Attractors of non-autonomous reaction-diffusion equations in \(L^p\)
cites work: A result on the existence of global attractors for semigroups of closed operators
cites work: Attractors for reaction-diffusion equations: existence and estimate of their dimension
cites work: Q5890172
DOI: 10.1016/J.NA.2009.02.083
description: scientific article; zbMATH DE number 5577466