Analysis & PDE

  • Anal. PDE
  • Volume 6, Number 5 (2013), 1089-1119.

Stabilization for the semilinear wave equation with geometric control condition

Romain Joly and Camille Laurent

Full-text: Open access

Abstract

In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity due to Robbiano, Zuily, Tataru and Hörmander. Some other consequences are also given for the controllability and the existence of a compact attractor.

Dans cet article, on prouve la décroissance exponentielle de l’équation des ondes semilinéaires avec un amortissement actif dans une zone satisfaisant seulement la condition de contrôle géométrique. La nonlinéarité est supposée sous-critique, défocalisante et analytique. La principale nouveauté par rapport aux résultats précédents est la preuve d’un résultat de prolongement unique en grand temps pour une solution non amortie. L’idée est d’utiliser un effet régularisant asymptotique prouvé par Hale et Raugel dans le contexte des systèmes dynamiques. Ensuite, une fois l’analyticité en temps prouvée, on applique un théorème de prolongement unique avec analyticité partielle dû à Robbiano, Zuily, Tataru et Hörmander. Des applications à la contrôlabilité et à l’existence d’attracteur global compact pour l’équation des ondes sont aussi données.

Article information

Source
Anal. PDE, Volume 6, Number 5 (2013), 1089-1119.

Dates
Received: 9 May 2012
Revised: 17 August 2012
Accepted: 27 September 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731398

Digital Object Identifier
doi:10.2140/apde.2013.6.1089

Mathematical Reviews number (MathSciNet)
MR3125551

Zentralblatt MATH identifier
1329.35062

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx] 35B65: Smoothness and regularity of solutions 93D20: Asymptotic stability 35L71: Semilinear second-order hyperbolic equations
Secondary: 35B41: Attractors

Keywords
damped wave equation stabilization analyticity unique continuation property compact attractor

Citation

Joly, Romain; Laurent, Camille. Stabilization for the semilinear wave equation with geometric control condition. Anal. PDE 6 (2013), no. 5, 1089--1119. doi:10.2140/apde.2013.6.1089. https://projecteuclid.org/euclid.apde/1513731398


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