The energy decay problem for hyperbolic equations of second order with dissipative terms
type: Item
instance of: scholarly article
title: The energy decay problem for hyperbolic equations of second order with dissipative terms
zbMATH Open document ID: 0656.35075
DOI: 10.1016/0022-247X(87)90205-8
author: Hiroshi Uesaka
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: Let \[ L_ A[u]=\{D^ 2+(A(x)/t)D+B(x)-\sum^{n}_{i,j=1}\partial_ ia_{ij}(x)\partial_ j\}u \] and \((L_ A+f)[u]=L_ A[u]+f(u).\) A bounded or unbounded domain of \(R^ n\) is denoted by \(\Omega\), \(\partial \Omega\) is its boundary. The total energy of u is defined \[ E(t;A)=\int_{\Omega}\{| Du|^ 2+\sum a_{ij}(x)\partial_ iu\partial_ ju+B(x)u^ 2+2F(u)\}dx+\int_{\partial \Omega}\Gamma (u)dS, \] where \(\Gamma\) is a certain quadratic form derived from the boundary condition by integration by part. Here the author investigates the energy decay properties of initial- or boundary-value problems for hyperbolic partial differential equations of second order with dissipative term A(x)/t: \((L_ A+f)[u]=0.\) Then he obtains Theorem 1. Under the assumptions A(x) be non-negative with additional assumptions, for any \(t\geq T\) \(E(t;A)\leq C/t^{\mu}\) holds where \(\alpha =\inf A(x)\) and \(\mu =Min\{2,\alpha \}.\) Theorem 2. Under some suitable assumptions and inf B(x)\(>0\), then \(E(t;A)=C/t^{\alpha}\) for \(t\geq T\).
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 4071483
zbMATH Keywords: total energy
zbMATH Keywords: energy decay
zbMATH Keywords: initial- or boundary-value problems
zbMATH Keywords: dissipative term
reviewed by: Akira Tsutsumi
MaRDI profile type: Publication
cites work: On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems
cites work: The total energy decay of solutions for the wave equation with a dissipative term
cites work: Q3218513
cites work: On the decay of solutions for some nonlinear evolution equations of second order
cites work: Q3913170
description: scientific article; zbMATH DE number 4071483