Global solution of a nonlinear string equation
type: Item
instance of: scholarly article
title: Global solution of a nonlinear string equation
zbMATH Open document ID: 0837.35095
author: William G. Newman
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-10-31T00:00:00Z
review text: The author studies the nonlinear equation \[ w_{tt} (x, t)- \Biggl(\lambda+ {2\over \pi} \int^\pi_0 w^2_x(x, t)dx\Biggr) w_{xx}(x, t)= P(x)\tag{1} \] on the interval \(0\leq x\leq \pi\), with \(\lambda< 0\), given the boundary conditions \(w(0, t)= w(\pi, t)= 0\) and the initial conditions \(w(x, 0)= f(x)\), \(w_t(x, 0)= g(x)\). A solution in the form \(w(x, t)= \sum^\infty_{j= 1} T_j(t)\sin jx\) is searched, and the global existence of a weak solution with \(\lambda< 0\) and \(P\not\equiv 0\) is shown, provided the initial data are sufficiently close to the tensile equilibrium solution of (1). Finally, the author shows that only one such solution exists.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 74K05
Mathematics Subject Classification ID: 45K05
zbMATH DE Number: 811157
zbMATH Keywords: uniqueness
zbMATH Keywords: global existence
reviewed by: Silvio Nocilla
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1198
OpenAlex ID: W1968960125
DOI: 10.1006/JMAA.1995.1198
description: scientific article; zbMATH DE number 811157