Lower bound of the time of existence for the nonlinear Klein-Gordon equation in one space dimension
type: Item
instance of: scholarly article
title: Lower bound of the time of existence for the nonlinear Klein-Gordon equation in one space dimension
zbMATH Open document ID: 0937.35160
DOI: 10.1016/S0294-1449(99)80028-6
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1999-11-21T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78475
review text: Let \(u\) be a solution to the semilinear Klein-Gordon equation in one space dimension, \[ \partial^2_tu-\partial^2_x u+u=F(u, \partial_tu, \partial_xu, \partial_t \partial_xu, \partial^2_xu) \] where \(F\) is a quadratic nonlinearity and Cauchy data \(u(t=0)=\varepsilon u_0\), \(\partial_t u(t=0)= \varepsilon u\), are small in \(C_0^\infty\). The aim of the paper is to obtain an explicit lower bound of the existence time of the solution \((T)\): \[ \lim\inf_{\varepsilon\to 0}\varepsilon^2\log T_\varepsilon \geq A. \] This is an improvement of the known result for \(A=0\). The expression for \(A\) is directly computed from the Cauchy data and the given nonlinearity. The author first constructs the asymptotic solution and then derives the approximate solution. An interesting constraint on the nonlinearity (called null-condition) is obtained for which \(A\) is \(\infty\).
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 1339269
zbMATH Keywords: semilinear Klein-Gordon
zbMATH Keywords: quadratic nonlinearity
zbMATH Keywords: Cauchy data
zbMATH Keywords: existence time
zbMATH Keywords: asymptotic solution
zbMATH Keywords: approximate solution
author: Jean-Marc Delort
MaRDI profile type: Publication
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OpenAlex ID: W2025949840
description: scientific article; zbMATH DE number 1339269