Attractors for damped nonlinear hyperbolic equations
type: Item
instance of: scholarly article
title: Attractors for damped nonlinear hyperbolic equations
zbMATH Open document ID: 0572.35071
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1987-01-01T00:00:00Z
review text: This work is devoted to the study of the longtime behavior of damped hyperbolic equations. We show that this behavior is finite dimensional by proving that the attractors associated with these equations have finite fractal dimension. This last result is based on estimates of the Lyapunov exponents on the attractors. In the first part we make some remarks and give an abstract setting that contains, in particular, the sine-Gordon equation and the nonlinear wave equation of quantum mechanics. In the second part we investigate the properties of the attractors including their dimension. And finally in the third part we give several examples and deal with a generalization to non-autonomous systems. In this last section we give a bound totaly explicit (in terms of the physical quantities) for the dimension of the maxmal attractor associated to the sine-Gordon equation with a forcing term which oscillates periodically in time.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3912893
zbMATH Keywords: oscillates periodically
zbMATH Keywords: longtime behavior
zbMATH Keywords: damped hyperbolic equations
zbMATH Keywords: finite dimensional
zbMATH Keywords: attractors
zbMATH Keywords: finite fractal dimension
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: sine-Gordon equation
zbMATH Keywords: nonlinear wave equation
zbMATH Keywords: quantum mechanics
author: Jean-Michel Ghidaglia
author: R. Temam
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3912893