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# Mathematical Research Letters

## Volume 26 (2019)

### Number 4

### Localized energy for wave equations with degenerate trapping

Pages: 991 – 1025

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n4.a3

#### Authors

#### Abstract

Localized energy estimates have become a fundamental tool when studying wave equations in the presence of asymptotically flat background geometry. Trapped rays necessitate a loss when compared to the estimate on Minkowski space. A loss of regularity is a common way to incorporate such. When trapping is sufficiently weak, a logarithmic loss of regularity suffices. Here, by studying a warped product manifold introduced by Christianson and Wunsch, we encounter the first explicit example of a situation where an estimate with an algebraic loss of regularity exists and this loss is sharp. Due to the global-in-time nature of the estimate for the wave equation, the situation is more complicated than for the Schrödinger equation. An initial estimate with sub-optimal loss is first obtained, where extra care is required due to the low frequency contributions. An improved estimate is then established using energy functionals that are inspired by WKB analysis. Finally, it is shown that the loss cannot be improved by any power by saturating the estimate with a quasimode.

Received 18 December 2017

Accepted 12 July 2018

Published 25 October 2019