Singular limit of some quasilinear wave equations with damping terms
type: Item
instance of: scholarly article
title: Singular limit of some quasilinear wave equations with damping terms
zbMATH Open document ID: 0841.35012
author: Tokio Matsuyama
published in: Advances in Differential Equations
publication date: 1996-07-18T00:00:00Z
review text: Let \(\Omega\subset \mathbb{R}^3\) be a bounded domain with smooth boundary \(\partial\Omega\). The author considers the mixed problem for the quasilinear wave equation \[ \varepsilon u_{tt}- \Biggl(\alpha+ 2\beta \int_\Omega |\nabla u(t, y)|^2 dy\Biggr) \Delta u+ \delta u_t= \mu u^3,\quad t> 0,\quad x\in \Omega, \] \[ u(0, x)= u_{0\varepsilon}(x),\;\partial_t u(0, x)= u_{1\varepsilon}(x),\quad x\in \Omega,\quad u(t, x)|_{\partial\Omega}= 0,\quad t\geq 0. \] Here \(\varepsilon\) is a positive parameter with \(0< \varepsilon\leq 1\). \(\alpha> 0\), \(\beta\geq 0\), \(\delta> 0\) and \(\mu\) are given constants, \(u= u(t, x)\) is an unknown real-valued function on \([0, \infty)\times \Omega\). The author considers a relation between the mixed problem and the reduced problem of the parabolic equation for \(\varepsilon= 0\). It is shown that the solution \(u_\varepsilon\) of the mixed problem converges, uniformly on any finite time interval, to the solution of the parabolic equation in an appropriate Hilbert space as \(\varepsilon\to 0\). For the proof several \(\varepsilon\) weighted energy estimates are obtained in order to evaluate the difference norm of \(u_\varepsilon- u\).
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 844387
zbMATH Keywords: singular limit
zbMATH Keywords: quasilinear wave equation
reviewed by: Akira Tsutsumi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 844387