An elementary proof of global or almost global existence for quasi-linear wave equations
type: Item
instance of: scholarly article
title: An elementary proof of global or almost global existence for quasi-linear wave equations
zbMATH Open document ID: 1056.35119
author: Kunio Hidano
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2004-11-05T00:00:00Z
review text: This paper gives a detailed outline on the energy integral method for proving global and almost global existence of solutions of nonlinear wave equations, methods which were developed mainly by Klainerman and Sideris. The main purpose of the author is to avoid the use of Lorentz boosts, and of direct estimates of fundamental solutions. The main result is an almost global existence theorem for solutions of quasilinear \(3+1\) dimensional wave equatons, together with an explicit estimate of the life span of solutions depending exponentially on the smallness parameter of the initial data; smallness is measured in generalized energies. The proof uses Sobolev type estimates introduced by Sideris, Hidano, and Nakanishi, which are applied in the proof of the main weighted state space estimates of Klainerman and Sideris, and in the proof of the energy estimates.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2114246
zbMATH Keywords: energy integral method
zbMATH Keywords: estimate of the life span
reviewed by: Horst Lange
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113246554
OpenAlex ID: W2112394093
cites work: Global existence of nonlinear elastic waves
cites work: Q4224971
cites work: Q4003851
cites work: Q4824303
cites work: Scattering problem for the nonlinear wave equation in the finite energy and conformal charge space
cites work: The global existence theorem for quasi-linear wave equations with multiple speeds
cites work: Global Small Amplitude Solutions of Nonlinear Hyperbolic Systems with a Critical Exponent under the Null Condition
cites work: Blow-up for quasi-linear wave equations in three space dimensions
cites work: Almost global existence of elastic waves of finite amplitude arising from small initial disturbances
cites work: Q3998613
cites work: Almost global existence to nonlinear wave equations in three space dimensions
cites work: Almost global existence for some semilinear wave equations
cites work: Almost global existence for quasilinear wave equations in three space dimensions
cites work: Uniform decay estimates and the lorentz invariance of the classical wave equation
cites work: Q3733338
cites work: Remarks on the global sobolev inequalities in the minkowski space Rn+1
cites work: Q4884687
cites work: Resonance-type behaviour in a system of nonlinear wave equations
cites work: Q4827074
cites work: Unique global existence and asymptotic behaviour of solutions for wave equations with non-coercive critical nonlinearity
cites work: Q4003172
cites work: The null condition and global existence of nonlinear elastic waves
cites work: Nonresonance and global existence of prestressed nonlinear elastic waves
cites work: Global existence for systems of nonlinear wave equations in 3D with multiple speeds
cites work: Q5475904
cites work: Q4462264
cites work: Q3995410
cites work: Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions
DOI: 10.2748/TMJ/1113246554
description: scientific article; zbMATH DE number 2114246