Local energy decay for several evolution equations on asymptotically euclidean manifolds

Bony, Jean-François; Häfner, Dietrich

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 2, pp. 311-335.

Soit $P$ une perturbation métrique à longue portée du laplacien euclidien sur ${ℝ}^d$, $d\ge 2$. On montre la décroissance de l’énergie locale des solutions des équations des ondes, de Klein-Gordon et de Schrödinger associées à $P$. Le problème est décomposé en une analyse basses et hautes fréquences. Afin de traiter les hautes fréquences, on fait une hypothèse de non capture. Pour les basses (resp. hautes) fréquences, on obtient un résultat général sur la décroissance de l’énergie locale pour le groupe $e^{itf\left(P\right)}$ où $f$ a un comportement prescrit en zéro (resp. à l’infini).

Let $P$ be a long range metric perturbation of the Euclidean Laplacian on ${ℝ}^d$, $d\ge 2$. We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to $P$. The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group $e^{itf\left(P\right)}$ where $f$ has a suitable development at zero (resp. infinity).

Classification : 35L05, 35J10, 35P25, 58J45, 81U30
Keywords: local energy decay, low frequencies, asymptotically euclidean manifolds, Mourre theory
Mot clés : décroissance de l'énergie locale, basses fréquences, variétés asymptotiquement euclidiennes, théorie de Mourre

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