A class of dissipative wave equations with time-dependent speed and damping
type: Item
instance of: scholarly article
title: A class of dissipative wave equations with time-dependent speed and damping
zbMATH Open document ID: 1261.35084
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-01-14T00:00:00Z
review text: The initial value problem to the equation \[ u_{tt}-\lambda (t)^2\Delta u+b(t)u_t+\lambda (t)\sum_{j=1}^nb_j(t)u_{x_j}+e(t)u=0\;\text{ in } (0+\infty)\times \mathbb{R}^n \] is considered. Under some assumptions on the coefficients of the equation the authors prove the energy estimate \[ \| u_t(t,\cdot)\| _{L_2}^2+\lambda (t)^2\| \nabla u_t(t,\cdot )\| _{L_2}^2\leq C\lambda (t)\exp(-\int_0^tb(s)\text{d}s)\left(\| u(0,\cdot )\|_{H^1}^2+\| u_t(0,\cdot )\| _{L_2}^2\right). \]
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 6125387
zbMATH Keywords: dissipative effects
zbMATH Keywords: energy estimate
zbMATH Keywords: dissipative wave equation
zbMATH Keywords: asymptotic behavior
author: M. D'Abbicco
author: M. R. Ebert
reviewed by: Marie Kopáčková
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.10.017
OpenAlex ID: W2023592919
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DOI: 10.1016/J.JMAA.2012.10.017
description: scientific article; zbMATH DE number 6125387