On channels of energy for the radial linearised energy critical wave equation in the degenerate case

Abstract

Channels of energy estimates control the energy of an initial data from that which it radiates outside a light cone. For the linearised energy critical wave equation they have been obtained in the radial case in odd dimensions, first in $3$ dimensions by Duyckaerts, Kenig and Merle (Camb. J. Math., 2013), then for general odd dimensions by the same authors (Comm. Math. Phys., 2020). We consider even dimensions, for which such estimates are known to fail (Côte, Kenig and Schlag, Math. Ann., 2014). We propose a weaker version of these estimates, around a single ground state as well as around a multisoliton. This allows us to prove the soliton resolution conjecture in six dimensions (Collot, Duyckaerts, Kenig and Merle, arXiv preprint 2201.01848, 2022 versions 1 and 2).

arXiv comment36 pages. This paper is an extension of the results of Section 3 of arXiv preprint 2201.01848, version 1. The main result of this paper is then used in arXiv preprint 2201.01848, version 2, where the former Section 3 of version 1 has been removed. arXiv admin note: text overlap with arXiv:2201.01848