Dispersive limits in the homogenization of the wave equation.
type: Item
instance of: scholarly article
title: Dispersive limits in the homogenization of the wave equation.
zbMATH Open document ID: 1070.35006
author: Grégoire Allaire
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2005-01-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73610
review text: The author studies a scalar wave equation (the Klein-Gordon equation) with a large potential in a periodic medium. The period is supposed of order \(\varepsilon\) and the potential is of order \(\varepsilon^{-2}\). The homogenized limit depends on the sign of the first cell eingenvalue \(\lambda_1\). If \(\lambda_1=0\), then the homogenized problem gives a standard wave equation. If \(\lambda_1\neq 0\), then, upon changing the time scale to focus on large times of order \(\varepsilon^{-1}\), the author obtains dispersive homogenized problems, i.e. equations which are not of second order in time. If \(\lambda_1 < 0\), the homogenized equation becomes parabolic, while, for \(\lambda_1 > 0\), the homogenized equation is of Schrödinger type.
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 2124646
zbMATH Keywords: homogenized problem
zbMATH Keywords: Klein-Gordon equation
zbMATH Keywords: periodic medium
reviewed by: Marco Codegone
MaRDI profile type: Publication
OpenAlex ID: W2332382308
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DOI: 10.5802/AFST.1055
description: scientific article; zbMATH DE number 2124646