Sharp polynomial energy decay for locally undamped waves
Matthieu Léautaud1; Nicolas Lerner2
1 Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Paris Diderot (Paris VII) Bâtiment Sophie Germain 75205 Paris Cedex 13 France
2 Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Pierre et Marie Curie (Paris VI) 4 Place Jussieu 75252 Paris cedex 05 France
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk 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.

DOI: 10.5802/slsedp.79
Matthieu Léautaud 1; Nicolas Lerner 2

1 Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Paris Diderot (Paris VII) Bâtiment Sophie Germain 75205 Paris Cedex 13 France
2 Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Pierre et Marie Curie (Paris VI) 4 Place Jussieu 75252 Paris cedex 05 France 
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     title = {Sharp polynomial energy decay for locally undamped waves},
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     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
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Matthieu Léautaud; Nicolas Lerner. Sharp polynomial energy decay for locally undamped waves. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 21, 13 p. doi : 10.5802/slsedp.79. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.79/

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