Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric conditions

Rayan Nasser; Nahla Noun; Ali Wehbe

doi:10.1016/j.crma.2019.01.005

Partial differential equations/Optimal control

Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric conditions
[Stabilisation d'une équation des ondes avec un amortissement de type Kelvin–Voigt localisé sous conditions géométriques optimales]

Présenté par : the Editorial Board

Rayan Nasser ^1,² ; Nahla Noun ³ ; Ali Wehbe ³

¹ Université libanaise, EDST & Hadath, Beyrouth, Liban
² Université de Bretagne occidentale, France
³ Université libanaise, Faculté des sciences 1 et EDST & Hadath, Beyrouth, Liban

Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 272-277.

Résumé
Abstract

Nous nous intéressons à l'étude de la stabilisation d'une équation des ondes avec un amortissement de type Kelvin–Voigt dans un domaine borné. L'amortissement est localisé via un coefficient singulier dans une partie du domaine. Nous montrons un résultat de stabilisation polynomiale en toute dimension d'espace dès que la région d'amortissement satisfait certaines conditions géométriques. La principale nouveauté de cette note est que les situations géométriques couvertes ici sont plus riches que celles considérées dans [25], [22], [16] et incluent notamment un exemple où la région d'amortissement n'est pas localisée dans un voisinage de la totalité ou d'une partie de la frontière.

The purpose of this note is to investigate the stabilization of the wave equation with Kelvin–Voigt damping in a bounded domain. Damping is localized via a non-smooth coefficient in a suitable subdomain. We prove a polynomial stability result in any space dimension, provided that the damping region satisfies some geometric conditions. The main novelty of this note is that the geometric situations covered here are richer than that considered in [25], [22], [16] and include in particular an example where the damping region is not localized in a neighborhood of the whole or a part of the boundary.

Reçu le : 2018-10-06
Accepté le : 2019-01-21
Publié le : 2019-03-01

DOI : 10.1016/j.crma.2019.01.005

Affiliations des auteurs :

Rayan Nasser ^{1, 2} ; Nahla Noun ³ ; Ali Wehbe ³

¹ Université libanaise, EDST & Hadath, Beyrouth, Liban
² Université de Bretagne occidentale, France
³ Université libanaise, Faculté des sciences 1 et EDST & Hadath, Beyrouth, Liban

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     author = {Rayan Nasser and Nahla Noun and Ali Wehbe},
     title = {Stabilization of the wave equations with localized {Kelvin{\textendash}Voigt} type damping under optimal geometric conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {272--277},
     publisher = {Elsevier},
     volume = {357},
     number = {3},
     year = {2019},
     doi = {10.1016/j.crma.2019.01.005},
     language = {en},
}

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Rayan Nasser; Nahla Noun; Ali Wehbe. Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric conditions. Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 272-277. doi : 10.1016/j.crma.2019.01.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.01.005/

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