On space-time means and strong global solutions of nonlinear hyperbolic equations
type: Item
instance of: scholarly article
title: On space-time means and strong global solutions of nonlinear hyperbolic equations
zbMATH Open document ID: 0705.35082
DOI: 10.1007/BF01161993
author: Philip Brenner
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174033
review text: The author studies the following nonlinear problem: \[ (*)\quad \partial^ 2_ tu-\Delta u+b(x,D)u+f(u)=0,\quad (t,x)\in {\mathbb{R}}_+\times {\mathbb{R}}^ n, \] \[ u(0,x)=\phi_ 0(x),\quad (\partial /\partial t)u(0,x)=\phi_ 1(x),\quad x\in {\mathbb{R}}^ n, \] where b(x,D) is any first order differential operator on \({\mathbb{R}}^ n\) with bounded smooth coefficients and f is a real function defined in \({\mathbb{R}}\). Under the hypothesis \(\phi_ 0\in L^ 2_ 2({\mathbb{R}}^ n)\), \(\phi_ 1\in L^ 1_ 2({\mathbb{R}}^ n)\) and \(f\in C^ 2\), \(f(0)=0\), \(F(\sigma)=\int^{\sigma}_{0}f(u)du\geq 0\), \(\forall \sigma \in {\mathbb{R}}\), \[ | f'(u)| \leq C(1+| u|)^{\rho -1},\quad 1<\rho <(n+2)/(n-3),\quad n\geq 3, \] he shows that there exists a unique solution u of problem (*) belonging to \(L_{\infty}^{loc}({\mathbb{R}};L^ 2_ 2({\mathbb{R}}^ n))\). For that, first he obtains the same result for a nonlinear Klein-Gordon equation with nonlinear term f(u) and data \(\phi_ 0\), \(\phi_ 1\). Then he applies this result to the general case (*). In his approach he uses properties of the solution of the corresponding linear Klein-Gordon equation and appropriate estimates for the nonlinear term.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35Q40
zbMATH DE Number: 4157086
zbMATH Keywords: strong global solutions
zbMATH Keywords: Klein-Gordon equation
reviewed by: M. Milla Miranda
MaRDI profile type: Publication
cites work: Global classical solutions of nonlinear wave equations
cites work: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
cites work: Space-time decay for solutions of wave equations
description: scientific article; zbMATH DE number 4157086