Uniform decay rates and attractors for evolution PDE's with boundary dissipation
type: Item
instance of: scholarly article
title: Uniform decay rates and attractors for evolution PDE's with boundary dissipation
zbMATH Open document ID: 0831.35022
published in: Journal of Differential Equations
publication date: 1996-02-22T00:00:00Z
review text: The main goal of the paper is to study the long time behaviour of solutions to some evolutionary equations with dissipative boundary conditions. A typical example of such a problem is the semilinear wave equation of the form \(u_{tt} - \Delta u + f(u) = 0\) in \(\Omega \subset \mathbb{R}^N\) complemented by the boundary conditions \({\partial u / \partial \nu} + u_t = 0\) on \(\partial \Omega\). Under various growth conditions concerning the function \(f\), the existence of a compact attractor or, if \(f\) is coercive, the convergence to zero of solutions for large times is established. The case of the so-called critical exponent is included. A related problem for the Schrödinger equation is also discussed. The proof is based on an observability lemma for a general linear operator based on Carleman type estimates. Various possibilities of obtaining such a result and possible generalizations for specific equations are discussed.
reviewed by: Eduard Feireisl
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 800399
zbMATH Keywords: dissipative boundary conditions
zbMATH Keywords: attractor
zbMATH Keywords: observability lemma
zbMATH Keywords: Carleman type estimates
author: Daniel Tataru
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1119
OpenAlex ID: W2083974094
DOI: 10.1006/JDEQ.1995.1119
description: scientific article; zbMATH DE number 800399