Stable periods of a semilinear wave equation and bifurcation of periodic solutions
type: Item
instance of: scholarly article
title: Stable periods of a semilinear wave equation and bifurcation of periodic solutions
zbMATH Open document ID: 0629.35013
DOI: 10.1007/BF00945406
author: P. Kötzner
author: Hansjörg Kielhöfer
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1987-01-01T00:00:00Z
review text: A semilinear wave equation \[ u_{tt}-u_{xx}=f(\lambda,x,u),\quad t\in {\mathbb{R}},\quad x\in (0,\pi),\quad \lambda \in {\mathbb{R}} \] with Dirichlet or Neumann boundary conditions is studied. It is supposed that f is a smooth function, \(f(\lambda,x,0)=0\), \(f_ u(\lambda,x,0)=a(\lambda)\) (with a(\(\lambda)\) independent of x) for all \(\lambda\in {\mathbb{R}}\), \(x\in <0,\pi >\). The linearized problem has nontrivial periodic solutions (in t) for any \(\lambda\in {\mathbb{R}}\). The question is whether some of these solutions persist in a perturbed form for the nonlinear problem. A set \(\Lambda\) dense in \({\mathbb{R}}\) is introduced such that for any \(a(\lambda_ 0)=a_ 0\in \Lambda\) there are finitely many periods \(P_ r(a_ 0)\) such that nontrivial \(P_ r(a_ 0)\)-periodic solutions bifurcate at \(\lambda_ 0\) from the trivial solution.
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35B20
zbMATH DE Number: 4023812
zbMATH Keywords: bifurcation of periodic solutions
zbMATH Keywords: semilinear
zbMATH Keywords: Dirichlet
zbMATH Keywords: Neumann
zbMATH Keywords: linearized problem
zbMATH Keywords: perturbed form
reviewed by: Milan Kučera
MaRDI profile type: Publication
cites work: Bifurcation of periodic solutions for a semilinear wave equation
cites work: A bifurcation theorem for critical points of variational problems
full work available at URL: https://doi.org/10.1007/bf00945406
OpenAlex ID: W2009454734
description: scientific article; zbMATH DE number 4023812