Global well-posedness for the exterior initial-boundary value problem to the Kirchhoff equation
type: Item
instance of: scholarly article
title: Global well-posedness for the exterior initial-boundary value problem to the Kirchhoff equation
zbMATH Open document ID: 1223.35238
author: Tokio Matsuyama
published in: Journal of the Mathematical Society of Japan
publication date: 2011-01-12T00:00:00Z
review text: The Kirchhoff equation describes the oscillations of initially stretched elastic strings. The displacement of the string thus depends only one space variable and a time variable. It satisfies a nonlinear second-order hyperbolic integro-differential equation. The author considers an \(n\)-dimensional generalization of this equation defined over an exterior open bounded domain in \(n\)-dimensional Euclidean space. It is assumed that the displacement vanishes at the boundary of the domain. The author proves by resorting to rather ingenious functional analysis techniques that the Kirchhoff equation admits a unique global solution provided that the initial values belong to certain Sobolev spaces of fractional order and satisfy some norm inequalities. The properties of the linear wave equation, generalized Fourier transform and the resolvent operator are amply used in the analysis.
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35R09
zbMATH DE Number: 5835139
zbMATH Keywords: generalized Fourier transform
zbMATH Keywords: scattering theory
reviewed by: Erdoğan S. Şuhubi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06241167
OpenAlex ID: W1971557906
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DOI: 10.2969/JMSJ/06241167
description: scientific article; zbMATH DE number 5835139