We consider the wave equation on a product cone and find a joint asymptotic expansion for forward solutions near null and future infinities. The rates of decay seen in the expansion are the resonances of a hyperbolic cone on the northern cap of the compactification and were computed by the authors in a previous paper. The expansion treats an asymptotic regime not considered in the influential work of Cheeger and Taylor. The main result follows the blueprint laid out in the asymptotically Minkowski setting; the key new element consists of propagation estimates near the conic singularities. The proof of the propagation estimates builds on the work of Melrose--Vasy--Wunsch in the spacetime and on Gannot--Wunsch in the semiclassical regime.
Download