Energy decay for the Klein–Gordon equation with highly oscillating damping

Royer, Julien

Annales Henri Lebesgue, Tome 1 (2018), pp. 297-312.

On s’intéresse dans cet article à l’équation de Klein–Gordon avec amortissement périodique. On montre sur un cas modèle que si la condition de contrôle géométrique usuelle est satisfaite, la décroissance de l’énergie est uniforme par rapport aux oscillations de l’amortissement, et en particulier la taille des dérivées ne joue aucun rôle. On montre également que sans cette condition géométrique la décroissance polynomiale de l’énergie est même un peu meilleure avec un amortissement fortement oscillant. Pour montrer ces estimées, on donne des versions dépendant d’un paramètre de résultats bien connus en théorie des semi-groupes.

We consider the free Klein–Gordon equation with periodic damping. We show on this simple model that if the usual geometric condition holds then the decay of the energy is uniform with respect to the oscillations of the damping, and in particular the sizes of the derivatives do not play any role. We also show that without geometric condition the polynomial decay of the energy is even slightly better for a highly oscillating damping. To prove these estimates we provide a parameter dependent version of well known results of semigroup theory.

Classification : 35L05, 35B40, 47B44, 47A10
Mots clés : Damped wave equation, energy decay, resolvent estimates, oscillating damping.

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