Global weak entropy solutions to quasilinear wave equations of Klein-Gordon and sine-Gordon type
type: Item
instance of: scholarly article
title: Global weak entropy solutions to quasilinear wave equations of Klein-Gordon and sine-Gordon type
zbMATH Open document ID: 0907.35084
author: Pierangelo Marcati
author: Roberto Natalini
published in: Journal of the Mathematical Society of Japan
publication date: 1999-02-15T00:00:00Z
review text: The objective of the paper is to prove the existence of global Lipschitz-continuous solutions to the Cauchy problem for the hyperbolic nonlinear partial differential equations of the following form: \[ w_{tt} -(\sigma (w_x))_x + f(w) = 0, \] where \(f\) is a smooth nonlinear function which is subject to the condition \(f(0) = 0\), and \(\sigma\) is another smooth function with a positive first derivative, satisfying an additional condition \(u\sigma'' (u) > 0\) for \(u\neq 0\). The existence is proved for all values of time, \(0 < t < +\infty\), and for the solutions on the whole axis, \(| x| < \infty\). For the proof, the equation is rewritten as a hyperbolic system. Then, an appropriate implicit finite-difference scheme for that system is introduced, which allows the authors to define some approximating solutions. The existence of the latter solution is established, and, finally, it is proved that the solution to the underlying equation also exists as a converging limit of the approximating solutions.
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1185908
zbMATH Keywords: global Lipschitz-continuous solutions
zbMATH Keywords: implicit finite-difference scheme
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05020433
OpenAlex ID: W2018504347
Wikidata QID: Q57777062
DOI: 10.2969/JMSJ/05020433
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description: scientific article; zbMATH DE number 1185908