Stabilization for the semilinear wave equation with geometric control condition

Romain Joly (1) , Camille Laurent (2)

1 ARTeHiS - Archéologie, Terre, Histoire, Sociétés [Dijon]
2 LJLL - Laboratoire Jacques-Louis Lions

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.

Mots clés: damped wave equation; stabilization; analyticity; unique continuation property; compact attractor

Domaines: Equations aux dérivées partielles [math.AP]

Romain Joly, Camille Laurent. Stabilization for the semilinear wave equation with geometric control condition. Analysis & PDE, 2013, 6 (5), pp.1089 - 1119. ⟨10.2140/apde.2013.6.1089⟩. ⟨hal-00696147v3⟩