Concentration close to the cone for linear waves

Abstract

We are concerned with solutions to the linear wave equation. Our main result concerns the computation of the asymptotic exterior energy outside of the cone $|x|\ge |t|+R$ for $R>0$ and odd dimension. This proves, in the general case, the results of Kenig–Lawrie–Liu–Schlag (2015) (which were restricted to radial data). Also, along the proof, we derive further expressions of the exterior energy (outside a shifted light cone), valid in all dimension and for non-radial data. In particular, we generalize the formulas of Côte–Kenig–Schlag (2014) obtained in the radial setting.