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Stabilization for the semilinear wave equation with geometric control condition

38 pages
Stabilization for the semilinear wave equation with geometric control condition Romain Joly? & Camille Laurent†‡ May 11, 2012 Abstract In this article, we prove the exponential stabilization of the semilinear wave equa- tion 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 Hormander. Some other consequences are also given for the controllability and the existence of a compact attractor. Key words: damped wave equation, stabilization, analyticity, unique continuation property, compact attractor. AMS subject classification: 35B40, 35B60, 35B65, 35L71, 93D20, 35B41. Resume Dans cet article, on prouve la decroissance exponentielle de l'equation des on- des semilineaires avec un amortissement actif dans une zone satisfaisant seulement la condition de controle geometrique. La nonlinearite est supposee sous-critique, defocalisante et analytique. La principale nouveaute par rapport aux resultats pre- cedents est la preuve d'un resultat de prolongement unique en grand temps pour une solution non amortie.

  • wave equation

  • topology generated

  • nonlinear wave

  • infinite time

  • whitney topology

  • time since

  • equation

  • decay can

  • equation des ondes


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