Concentration close to the cone for linear waves

TL;DR

This paper analyzes the asymptotic behavior of solutions to the linear wave equation, demonstrating energy concentration near the light cone and generalizing previous radial results to the non-radial case.

Contribution

It provides a new asymptotic formula for linear wave solutions using the Radon transform, extending prior radial results to the general case.

Findings

Energy concentrates near the light cone asymptotically.

Derived formulas for exterior energy outside a shifted light cone.

Extended results on energy discrepancy to non-radial initial data.

Abstract

We are concerned with solutions to the linear wave equation. We give an asymptotic formula for large time, valid in the energy space, via an operator related to the Radon transform. This allows us to show that the energy is concentrated near the light cone. This allows to derive further expressions the exterior energy (outside a shifted light cone). We in particular generalize the formulas of [CKS14] obtained in the radial setting. In odd dimension, we study the discrepancy of the exterior energy regarding initial energy, and prove in the general case the results of [KLLS15] (which were restricted to radial data).