\(L^{2}\)-estimates of solutions for damped wave equations with space-time dependent damping term
type: Item
instance of: scholarly article
title: \(L^{2}\)-estimates of solutions for damped wave equations with space-time dependent damping term
zbMATH Open document ID: 1184.35213
author: Jiayun Lin
author: Kenji Nishihara
author: Jian Zhai
published in: Journal of Differential Equations
publication date: 2010-01-05T00:00:00Z
review text: The Cauchy problem to the equation \(u_{tt}-\Delta u +b_0(1+\left|x\right|^2)^{-\alpha /2}(1+t)^{-\beta}u_t+\left|u\right|^{\rho -1}u=0\) in \(\mathbb R^{+}\times\mathbb R^n\), where \(\alpha ,\beta \geq 0,\, \alpha+\beta <1\), is considered. Using the local existence theorem and the weighted energy method the authors prove global existence and \(L^2\)-decay rate of the solution. Throughout the paper the exponent \(\rho \) yields: \(1<\rho <\infty \) for \(n=1,2\) and \(1<\rho \leq \frac{n}{n-2}\) for \(n\geq 3\).
reviewed by: Marie Kopáčková
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 5653494
zbMATH Keywords: weighted energy method
zbMATH Keywords: decay rate
MaRDI profile type: Publication
full work available at URL: j.jde.2009.09.022
OpenAlex ID: W2037196872
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DOI: 10.1016/J.JDE.2009.09.022
description: scientific article; zbMATH DE number 5653494