Large time behavior of solutions for a system of nonlinear damped wave equations
type: Item
instance of: scholarly article
title: Large time behavior of solutions for a system of nonlinear damped wave equations
zbMATH Open document ID: 1233.35038
author: Takayoshi Ogawa
author: H. Takeda
published in: Journal of Differential Equations
publication date: 2011-10-21T00:00:00Z
review text: This work is concerned with semilinear wave equations having linear dampings in the \(n\)-dimensional Euclidean space. The dependent variable is vector-valued with \(m\) components, \(m\) is assumed to be no less than 2. Each component of the nonlinear coupling function depending only on the dependent variable is bounded above by a power-type function prescribed by the product of the absolute values of the components of the dependent variable with given exponents. A similar bound does exist for the absolute value of the difference of of coupling functions for two different arguments. Initial values of the dependent variable and its time derivative at \(t=0\) are prescribed vector-valued functions. The authors show for \(n=1,2,3\) that if the sum of the exponents satisfies a certain inequality and the sum of the norms of initial values in appropriate Sobolev and Lebesgue spaces is sufficiently small, then there exists a unique global solution satisfying an optimal uniform decay rate. An asymptotic profile for large time is also obtained. The proof is based on the previously obtained decomposition of fundamental solutions of the linear damped wave equation into dissipative and hyperbolic parts.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35L51
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 5961149
zbMATH Keywords: decay estimate
zbMATH Keywords: asymptotic profile
zbMATH Keywords: decomposition of fundamental solutions
reviewed by: Erdoğan S. Şuhubi
MaRDI profile type: Publication
full work available at URL: j.jde.2011.07.034
OpenAlex ID: W2041159535
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DOI: 10.1016/J.JDE.2011.07.034
description: scientific article; zbMATH DE number 5961149