A remark on the stabilization of the 1-d wave equation

Serge Nicaise; Julie Valein

doi:10.1016/j.crma.2009.11.015

Partial Differential Equations

A remark on the stabilization of the 1-d wave equation

Presented by: Gilles Lebeau

Serge Nicaise ¹; Julie Valein ²

¹ Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes cedex 9, France
² Institut Elie Cartan Nancy (IECN), Nancy-université & INRIA (Project-Team CORIDA), 54506 Vandoeuvre-lès-Nancy cedex, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 47-51.

Abstract
Résumé

We consider the wave equation on an interval of length 1 with an interior damping at ξ and with Dirichlet boundary condition at the two ends. It is well known that, if ξ is rational, the energy does not decay to 0. In this case, we prove that the energy decays exponentially to a constant which we identify.

Nous considérons l'équation des ondes sur un intervalle de longueur 1 avec un amortissement en un point ξ intérieur et avec la condition au bord de Dirichlet aux deux extrémités. Il est bien-connu que, si ξ est rationnel, l'énergie ne tend pas vers 0. Dans ce cas, nous prouvons que l'énergie décroît exponentiellement vers une constante que l'on explicitera.

Received: 2009-06-23
Accepted: 2009-11-25
Published online: 2009-12-24

DOI: 10.1016/j.crma.2009.11.015

Author's affiliations:

Serge Nicaise ¹; Julie Valein ²

¹ Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes cedex 9, France
² Institut Elie Cartan Nancy (IECN), Nancy-université & INRIA (Project-Team CORIDA), 54506 Vandoeuvre-lès-Nancy cedex, France

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Serge Nicaise; Julie Valein. A remark on the stabilization of the 1-d wave equation. Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 47-51. doi : 10.1016/j.crma.2009.11.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.11.015/

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