Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domain

TL;DR

This paper establishes global dispersive estimates for wave and Klein-Gordon equations in the Friedlander domain, highlighting differences in decay rates at low frequencies compared to flat space.

Contribution

It provides the first detailed analysis of dispersive behavior inside the Friedlander domain, emphasizing the role of caustics and frequency regimes.

Findings

Dispersive decay is proven globally in time for both equations.

Klein-Gordon exhibits worse decay than the wave at low frequencies.

Significant differences between curved and flat space dispersive behavior are identified.

Abstract

We prove global in time dispersion for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. Moreover, we uncover a significant difference between Klein-Gordon and the wave equation in the low frequency, large time regime, where Klein-Gordon exhibits a worse decay that the wave, unlike in the flat space.