Sharp polynomial energy decay for locally undamped waves
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 21, 13 p.

In this note, we present the results of the article [LL14], and provide a complete proof in a simple case. We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in traversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function.

@article{SLSEDP_2014-2015____A21_0,
     author = {L\'eautaud, Matthieu and Lerner, Nicolas},
     title = {Sharp polynomial energy decay for locally undamped waves},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:21},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2014-2015},
     doi = {10.5802/slsedp.79},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/slsedp.79/}
}
Léautaud, Matthieu; Lerner, Nicolas. Sharp polynomial energy decay for locally undamped waves. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 21, 13 p. doi : 10.5802/slsedp.79. http://archive.numdam.org/articles/10.5802/slsedp.79/

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