Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary
type: Item
instance of: scholarly article
title: Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary
zbMATH Open document ID: 1429.35019
published in: Discrete and Continuous Dynamical Systems
publication date: 2019-06-28T00:00:00Z
review text: A family of IBV hyperbolic problems of the form \(u^\epsilon_{tt}+\frac{1}{\epsilon}\chi_{\omega_\epsilon}u^\epsilon_t-\Delta u^\epsilon+\lambda u^\epsilon= f_\epsilon+\frac{1}{\epsilon}\chi_{\omega_\epsilon}g_\epsilon\) in \(\Omega \times(0,T)\), \(\frac{\partial u^\epsilon}{\partial n} =0\) on \(\Gamma \times(0,T)\), \(u^\epsilon(0,x)= u^\epsilon_0(x)\), \(u^\epsilon_t(0,x)= v^\epsilon_0(x)\) in \(\Omega\) is considered. Here \(\Gamma\) is the boundary of \(\Omega\), and \(\omega_\epsilon\) denotes a neighborhood of \(\Gamma\) that converges to the boundary when \(\epsilon\) goes to zero, and \(\chi_{\omega_\epsilon}\) denotes its characteristic function. The goal of this work is to prove that, when \(\epsilon\to 0\), the family of solutions \(u^\epsilon\) converges to a limit function \(u^0\) which is the solution of the IBV hyperbolic problem with boundary feedback damping term: \(u^0_{tt}-\Delta u^0+\lambda u^0 =f\) in \(\Omega\times(0,T)\), \(u^0_t+\frac{\partial u^0}{\partial n}=g\) on \(\Gamma\times(0,T)\), \(u^0(0,x)=u_0(x)\), \(u^0_t(0,x)=v_0(x)\) in \(\Omega\). Initial values are limits of the initial data in the considered system. Well-posedness results for both systems are proved. Mild and strict solutions for both problems are built. Energy estimates are considered. Convergence of mild solutions as well as of strict solutions are investigated.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 7074655
zbMATH Keywords: localized damping
zbMATH Keywords: concentrating integrals
zbMATH Keywords: energy estimates
author: Angela Jiménez-Casas
author: Anibal Rodriguez-Bernal
MaRDI profile type: Publication
full work available at URL: dcds.2019208
OpenAlex ID: W2955219112
Wikidata QID: Q127746046
DOI: 10.3934/DCDS.2019208
description: scientific article; zbMATH DE number 7074655