The radiation field on product cones

Abstract

We consider the wave equation on a product cone and find a joint asymptotic expansion for solutions near null and future infinities. The rates of decay seen in the expansion at future infinity are the resonances of a hyperbolic cone and were computed by the authors in [2]. The expansion treats an asymptotic regime not considered in the influential work of Cheeger and Taylor [5,6]. The main result follows the blueprint laid out in the [3,4] with key new elements including propagation estimates near the conic singularities. The proof of the propagation estimates extends prior work of Melrose–Vasy–Wunsch [19] and Gannot–Wunsch [10].