Construction of periodic solutions of nonlinear wave equations in higher dimension
type: Item
instance of: scholarly article
title: Construction of periodic solutions of nonlinear wave equations in higher dimension
zbMATH Open document ID: 0834.35083
DOI: 10.1007/BF01902055
published in: Geometric and Functional Analysis. GAFA
publication date: 1996-04-01T00:00:00Z
full work available at URL: https://eudml.org/doc/58205
review text: The author deals with the existence of space and time periodic solutions of nonlinear wave equations \(u_{tt}- \Delta u+ F' (u)=0\). In particular, for the periodic wave equation in dimension \(d\) \[ u_{tt}- \Delta u+ \rho u+ u^3=0 \tag{1} \] the following proposition is demonstrated: Let \(\rho\in \mathbb{R}\) satisfy a condition of the form \[ \Biggl|\sum^r_{j=0} a_j \rho^j \Biggr|> \Bigl(\sum |a_j|\Bigr)^{- c(r)} \qquad \text{for all } \{a_j\}\in \mathbb{Z}^{r+1} \setminus \{0\}. \] Fix \(m_0\in \mathbb{Z}^d \setminus \{0\}\). Then there is a Cantor set \(C\) of positive measure in an interval \([0, \delta ]\) such that for \(p_0\in C\) there exists a solution of (1) of the form \(u= p_0 \cos (\langle m_0, x\rangle+ \lambda t)+ O(p^3_0)\), where \(\lambda^2= |m_0 |^2+ \rho+ {3\over 4} p^2_0+ O(p^4_0)\).
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 800441
zbMATH Keywords: nonlinear wave equations
zbMATH Keywords: solutions periodic in time and space
author: Jean Bourgain
reviewed by: Silvio Nocilla
MaRDI profile type: Publication
Wikidata QID: Q97672484
cites work: Q4763964
cites work: Newton's method and periodic solutions of nonlinear wave equations
cites work: Nearly integrable infinite-dimensional Hamiltonian systems
OpenAlex ID: W2041599937
description: scientific article; zbMATH DE number 800441