On classification of non-radiative solutions for various energy-critical wave equations

TL;DR

This paper classifies non-radiative solutions of energy-critical wave equations across various dimensions, extending previous results and establishing a unique maximal extension for these solutions.

Contribution

It generalizes the classification of non-radiative solutions to higher dimensions and establishes their unique maximal extension.

Findings

Classified asymptotic behavior of non-radiative solutions in various dimensions.

Identified a $k$-parameter family of solutions depending on the dimension.

Established a unique maximal extension for these solutions.

Abstract

Non-radiative solutions of energy critical wave equations are such that their energy in an exterior region $|x|>R+|t|$ vanishes asymptotically in both time directions. This notion, introduced by Duyckaerts, Kenig and Merle (J. Eur. Math. Soc., 2011), has been key in solving the soliton resolution conjecture for these equations in the radial case. In the present paper, we first classify their asymptotic behaviour at infinity, showing that they correspond to a $k$-parameters family of solutions where $k$ depends on the dimension. This generalises the previous results (Duyckaerts, Kenig and Merle, Camb. J. Math., 2013 and Duyckaerts, Kenig, Martel and Merle, Comm. Math. Phys., 2022) in three and four dimensions. We then establish a unique maximal extension of these solutions.