A singular perturbed problem for semilinear wave equations with small parameter
type: Item
instance of: scholarly article
title: A singular perturbed problem for semilinear wave equations with small parameter
zbMATH Open document ID: 0958.35010
author: Han Yang
published in: Discrete and Continuous Dynamical Systems
publication date: 2000-08-16T00:00:00Z
review text: The author considers the Cauchy problem: \(\delta u_{tt}-\Delta u+u_t= F(u,\nabla u)\), \(t=0:u= \varepsilon u_0(x)\), \(u_t=\varepsilon u_1(x)\), where \(\delta\) and \(\varepsilon\) are small positive parameters, \(\nabla\) is the gradient operator and \(\Delta u=\sum{\partial^2u \over\partial \chi^2_i}\), and \(F\) is a smooth function satisfying given conditions. The author uses the global iteration method to obtain a lower bound independent of \(\delta\) on the life span of the classical solutions to the above system, and compares the solution of the perturbed system to that of the unperturbed system i.e. whether \(\delta=0\) or \(\delta>0\). The author proves that with the exception for a small boundary layer, the solution of the system when \(\delta>0\) converges to the solution of the limit case, i.e. when \(\delta=0\). It is also shown, in a rigorous way, that the perturbation term \(\delta u_n\) has an appreciable effect only for a short time.
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 1492591
zbMATH Keywords: global iteration method
zbMATH Keywords: small boundary layer
reviewed by: Hussain S. Nur
MaRDI profile type: Publication
full work available at URL: dcds.1999.5.473
OpenAlex ID: W2331726643
DOI: 10.3934/DCDS.1999.5.473
description: scientific article; zbMATH DE number 1492591