# Local energy decay for scalar fields on time dependent non-trapping backgrounds

## TL;DR

This paper studies how scalar wave solutions decay locally over time on non-trapping, asymptotically flat space-times, providing spectral characterizations and decay estimates in stationary and nearly stationary cases.

## Contribution

It offers a comprehensive spectral analysis of local energy decay and introduces decay estimates for almost stationary and symmetric wave equations.

## Key findings

- Spectral characterization of local energy decay in stationary cases
- Two-point decay estimate for almost stationary wave equations
- Exponential trichotomy with stable and unstable subspaces

## Abstract

We consider local energy decay estimates for solutions to scalar wave equations on nontrapping asymptotically flat space-times. Our goals are two-fold. First we consider the stationary case, where we can provide a full spectral characterization of local energy decay bounds; this characterization simplifies in the stationary symmetric case. Then we consider the almost stationary, almost symmetric case. There we establish two main results: The first is a "two point" local energy decay estimate which is valid for a general class of (non-symmetric) almost stationary wave equations which satisfy a certain nonresonance property at zero frequency. The second result, which also requires the almost symmetry condition, is to establish an exponential trichotomy in the energy space via finite dimensional time dependent stable and unstable sub-spaces, with an infinite dimensional complement on which solutions disperse via the usual local energy decay estimate.