On étudie la stabilisation et le contrôle interne de l’équation de Klein-Gordon critique sur des variétés de dimension 3. Sous des conditions géométriques légèrement plus fortes que la condition de contrôle géométrique classique, on prouve la décroissance exponentielle de solutions bornées dans l’espace d’énergie mais petites dans des normes plus faibles. La preuve combine la décomposition en profils et des arguments microlocaux. Cette décomposition, analogue à celle de Bahouri-Gérard [2] sur , nécessite l’analyse de certains effets dus à la géométrie. Elle utilise des résultats de S. Ibrahim [16] sur le comportement d’ondes de concentration sur les variétés.
We study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri-Gérard [2] on , is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim [16] on the behavior of concentrating waves on manifolds.
@article{JEDP_2011____A6_0,
author = {Laurent, Camille},
title = {On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
year = {2011},
pages = {1-17},
doi = {10.5802/jedp.78},
language = {en},
url = {http://dml.mathdoc.fr/item/JEDP_2011____A6_0}
}
Laurent, Camille. On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold. Journées équations aux dérivées partielles, (2011), pp. 1-17. doi : 10.5802/jedp.78. http://gdmltest.u-ga.fr/item/JEDP_2011____A6_0/
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