Uniform estimates for solutions of nonlinear Klein-Gordon equations
type: Item
instance of: scholarly article
title: Uniform estimates for solutions of nonlinear Klein-Gordon equations
zbMATH Open document ID: 0568.35068
DOI: 10.1016/0022-1236(85)90057-6
author: Thierry Cazenave
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: The estimates derived in this paper concern the solutions of a nonlinear Klein-Gordon equation \(u_{tt}-\Delta u+mu=g(u)\) in \(\Omega\), \(u=0\) in \(\partial \Omega\), where \(\Omega\) is an open subset of \({\mathbb{R}}^ N\), \(m>0\), and g satisfies some growth conditions. The main result of this paper is that under some superlinearity condition on g (namely if \(xg(x)\geq (2+\epsilon)\int^{x}_{0}g(s)ds)\) then all global solutions are uniformly bounded in \(H^ 1_ 0(\Omega)\). Furthermore, for large time, the bound only depends on the energy of the initial data, that is, on the quantity \[ E(u_ 0,v_ 0)=\int_{\Omega}\{| v_ 0(x)|^ 2+| \nabla u_ 0(x)|^ 2+m/2| u_ 0(x)|^ 2-G(u_ 0(x))\}dx. \] In addition, solutions on the whole line, that is, solutions that exist for \(t\in]-\infty,+\infty [\), are uniformly bounded in terms of the energy of the initial data. As simple consequences we obtain conditions on the initial data under which the solution of (K.G) blows up in a finite time. Similar results were obtained for certain types of ordinary differential equation and for parabolic equations. Here the author demonstrates existence of such phenomena for the K-G equation and conjectures possible extensions.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 3907001
zbMATH Keywords: estimates
zbMATH Keywords: nonlinear Klein-Gordon equation
zbMATH Keywords: superlinearity condition
zbMATH Keywords: global solutions
zbMATH Keywords: uniformly bounded
zbMATH Keywords: energy of the initial data
zbMATH Keywords: blows up
reviewed by: Vadim Komkov
MaRDI profile type: Publication
full work available at URL: 0022-1236(85)90057-6
OpenAlex ID: W2038817278
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description: scientific article; zbMATH DE number 3907001