Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary

Ángela Jiménez-Casas; Aníbal Rodríguez-Bernal

doi:10.3934/dcds.2019208

Article Contents

2019, Volume 39, Issue 9: 5125-5147. Doi: 10.3934/dcds.2019208

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Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary

Ángela Jiménez-Casas^1, and
Aníbal Rodríguez-Bernal^2,3, ,

1.
Grupo de Dinámica No Lineal, Universidad Pontificia Comillas de Madrid, C/Alberto Aguilera 23, 28015 Madrid, Spain
2.
Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040, Madrid
3.
ICMAT^§, Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Spain

^* Corresponding author: A. Rodríguez–Bernal
^* Corresponding author: A. Rodríguez–Bernal

The first author is supported by Projects MTM2016-75465, MICINN and GR58/08 Grupo 920894, UCM, Spain and FIS2016-78883-C2-2-P(AEI/FEDER, U.E.), Spain.
The second author is supported by Projects MTM2016-75465, MICINN and GR58/08 Grupo 920894, UCM, Spain and EPSRC grant EP/R023778/1, UK.
ICMAT^§ is Partially supported by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO)

Received: May 31, 2018

Revised: January 31, 2019

Published: May 2019

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Abstract

In this paper we show how solutions of a wave equation with distributed damping near the boundary converge to solutions of a wave equation with boundary feedback damping. Sufficient conditions are given for the convergence of solutions to occur in the natural energy space.

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Mathematics Subject Classification: Primary: 35L05, 35L15, 35B25.

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