Acta Mathematica

Volume 230 (2023)

Number 1

Soliton resolution for the radial critical wave equation in all odd space dimensions

Pages: 1 – 92



Thomas Duyckaerts (Sorbonne Paris Cité LAGA (UMR CNRS 7539), Université Paris 13, Villetaneuse, France)

Carlos Kenig (Department of Mathematics, University of Chicago, Illinois, U.S.A.)

Frank Merle (Department of Mathematics, University of Cergy-Pontoise, Cergy, France)


Consider the energy-critical focusing wave equation in odd space dimension $N \geqslant 3$. The equation has a non-zero radial stationary solution $W$, which is unique up to scaling and sign change. In this paper we prove that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$’s, decoupled by the scaling, and a radiation term.

The proof essentially boils down to the fact that the equation does not have purely non-radiative multi-soliton solutions. The proof overcomes the fundamental obstruction for the extension of the 3D case (treated in [21]) by reducing the study of a multisoliton solution to a finite-dimensional system of ordinary differential equations on the modulation parameters. The key ingredient of the proof is to show that this system of equations creates some radiation, contradicting the existence of pure multi-solitons.


focusing wave equation, dynamics, soliton resolution, global solutions, blow-up

Carlos Kenig was partially supported by NSF Grants DMS-14363746 and DMS-1800082.

Received 18 December 2019

Accepted 23 May 2021

Published 24 March 2023