Energy decay for a degenerate hyperbolic equation with a dissipative term
type: Item
instance of: scholarly article
title: Energy decay for a degenerate hyperbolic equation with a dissipative term
zbMATH Open document ID: 0955.35049
author: Fumihiko Hirosawa
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2001-02-21T00:00:00Z
review text: The goal of the paper is to study decay properties of the total energy of solutions to hyperbolic equations. In opposite to other papers which prefer to study wave equations with mass and dissipation, the author is interested in the Cauchy problem \[ u_{tt}- a(t)\Delta u+ 2b(t) u_t+ m^2u= 0\quad\text{in }\mathbb{R}^{n+1}_+, \] \[ u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x)\quad\text{on }\mathbb{R}^n, \] in two situations: 1) the equation is strictly hyperbolic, \(a(t)\geq c>0\), it is enough to study the behaviour of the Sobolev-type energy; 2) the equation is weakly hyperbolic, \(a(t)\geq 0\), one cannot expect the existence of Sobolev solutions, the regularity \(a(t){\i}C^1[0,\infty)\) guarantees the existence of Gevrey solutions in Gevrey spaces \(s< 3/2\), therefore it is necessary to study the infinite-order energy. The main results of the paper yield an exponential decay of both energies. The method is based on an approach, which was introduced by Zuazua. The main problem is to estimate the partial energies and to understand the influence of oscillations of the leading coefficient \(a= a(t)\). It would be interesting to compare the results with results, which are obtained by an application of the method of representation of the solution by Fourier multipliers.
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35L80
zbMATH DE Number: 1439855
zbMATH Keywords: total energy
zbMATH Keywords: Sobolev-type energy
zbMATH Keywords: Gevrey solutions
zbMATH Keywords: infinite-order energy
zbMATH Keywords: exponential decay
reviewed by: Michael Reissig
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195143608
OpenAlex ID: W1996958936
cites work: Q3331627
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DOI: 10.2977/PRIMS/1195143608
description: scientific article; zbMATH DE number 1439855