# Dispersion for the wave equation outside a ball and counterexamples

## TL;DR

This paper investigates dispersive estimates for the wave equation outside a ball in various dimensions, showing standard estimates in three dimensions and dispersion losses in higher dimensions at the Poisson spot.

## Contribution

It establishes dispersive estimates outside a ball in 3D and demonstrates dispersion losses at the Poisson spot in dimensions four and higher.

## Key findings

- Dispersive estimates hold in 3D.
- Losses in dispersion occur at the Poisson spot in higher dimensions.
- Dispersion behavior varies with dimension.

## Abstract

The purpose of this note is to prove dispersive estimates for the wave equation outside a ball in R^d. If d = 3, we show that the linear flow satisfies the dispersive estimates as in R^3. In higher dimensions d $\ge$ 4 we show that losses in dispersion do appear and this happens at the Poisson spot.