On some eigenvalue problems related to the wave equation with indefinite damping
type: Item
instance of: scholarly article
title: On some eigenvalue problems related to the wave equation with indefinite damping
zbMATH Open document ID: 0857.35077
author: Pedro Freitas
published in: Journal of Differential Equations
publication date: 1997-03-09T00:00:00Z
review text: The author considers the damped wave equation of the form \[ u_{tt}+ \varepsilon a(x)u_t=\Delta u+b(x)u \qquad \text{for} \quad x\in\Omega, \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), and solutions are supposed to satisfy homogeneous Dirichlet boundary conditions. The novelty of the paper lies in the fact that the function \(a\) is allowed to change sign on \(\Omega\), specifically, the sets \(\Omega_+\), \(\Omega_-\) on which \(a\) is positive, negative respectively are of positive measure. The rather unexpected main result then states that the zero solution is unstable provided \(\varepsilon\geq \varepsilon_0\) is large enough. The proof is based on careful spectral analysis of the problem.
reviewed by: Eduard Feireisl
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35P05
zbMATH DE Number: 881712
zbMATH Keywords: indefinite damping
zbMATH Keywords: damped wave equation
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0072
OpenAlex ID: W1976489357
DOI: 10.1006/JDEQ.1996.0072
description: scientific article; zbMATH DE number 881712