Sharp resolvent and time decay estimates for dispersive equations on asymptotically Euclidean backgrounds

TL;DR

This paper develops a robust method to establish sharp time decay estimates for wave, Klein-Gordon, and Schrödinger equations on curved geometries, extending Euclidean decay results to asymptotically Euclidean backgrounds and boundary value problems.

Contribution

It introduces a general approach for proving optimal decay rates for dispersive PDEs on curved spaces, including boundary conditions, matching Euclidean decay behavior.

Findings

Sharp decay estimates for wave, Klein-Gordon, Schrödinger equations on curved backgrounds

Extension of decay results to boundary value problems

Method applicable under broad geometric assumptions

Abstract

The purpose of this article is twofold. First we give a very robust method for proving sharp time decay estimates for the most classical three models of dispersive Partial Differential Equations, the wave, Klein-Gordon and Schr{\"o}dinger equations, on curved geometries, showing under very general assumptions the exact same decay as for the Euclidean case. Then we also extend these decay properties to the case of boundary value problems.