Local energy decay for several evolution equations on asymptotically Euclidean manifolds. (Décroissance de l’énergie locale pour un certain nombre d’équations d’évolution sur des variétés asymptotiquement euclidiennes.) (English. French summary) Zbl 1263.58008
The paper is treating the local energy decay for several evolution equations associated to long range metric perturbations of the Euclidean Laplacian: the wave, Klein-Gordon and Schrödinger equations. The study is separated by the low and, respectively, high frequency analysis. In low (respectively high) frequency, denoting by \(P\) a long range perturbation of the Euclidean Laplacian, and assuming that for the high energy part that \(P\) is non-trapping, the authors obtain a general result about the local energy decay for the group \(e^{itf(P)}\) where \(f\) has a suitable development at zero (respectively at infinity).
Reviewer: Alina Stancu (Montréal)
MSC:
58J37 | Perturbations of PDEs on manifolds; asymptotics |
35B40 | Asymptotic behavior of solutions to PDEs |
35J15 | Second-order elliptic equations |
35R01 | PDEs on manifolds |