On the existence of multiple solutions for a nonlinear wave equation
type: Item
instance of: scholarly article
title: On the existence of multiple solutions for a nonlinear wave equation
zbMATH Open document ID: 0649.35061
DOI: 10.1016/0022-0396(88)90031-9
author: Norimichi Hirano
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: This paper is devoted to the existence of multiple time-periodic solutions for the problem \[ u_{tt}-u_{xx}=g(u)\text{ in }(0,\pi)\times {\mathbb{R}},\quad u(0,t)=u(\pi,t)=0\text{ for all } t\in {\mathbb{R}}, \] when g is continuous. The author shows that the number of weak solutions is not smaller than the number of those eigenvalues of \(\partial^ 2_ t-\partial^ 2_ x\) corresponding to the odd eigenfunctions and which are crossed by \(g(\xi)/\xi\) as \(\xi\) varies from 0 to \(\infty\). The proof is based on an existence theorem for a pseudo- monotone mapping.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 47H05
zbMATH DE Number: 4059967
zbMATH Keywords: existence
zbMATH Keywords: multiple time-periodic solutions
zbMATH Keywords: weak solutions
zbMATH Keywords: pseudo- monotone mapping
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90031-9
OpenAlex ID: W1989011560
cites work: Q3937768
cites work: Q4094029
cites work: Periodic solutions of nonlinear vibrating strings and duality principles
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cites work: Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces
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cites work: Existence Theorems on Unbounded Sets in Banach Spaces
description: scientific article; zbMATH DE number 4059967