Local energy decay for several evolution equations on asymptotically euclidean manifolds

TL;DR

This paper establishes local energy decay for wave, Klein-Gordon, and Schrödinger equations on asymptotically Euclidean manifolds with long-range perturbations, using frequency analysis and non-trapping conditions.

Contribution

It provides a unified approach to local energy decay for multiple evolution equations on asymptotically Euclidean manifolds, covering both low and high frequency regimes.

Findings

Proves local energy decay under non-trapping conditions for high energies.

Establishes decay results for low frequencies with suitable spectral assumptions.

Provides a general framework applicable to various evolution equations.

Abstract

Let P be a long range metric perturbation of the Euclidean Laplacian on R^d, d>1. We prove local energy decay for the solutions of the wave, Klein-Gordon and Schroedinger 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 exp(itf(P)) where f has a suitable development at zero (resp. infinity).