Soliton resolution for the radial critical wave equation in all odd space dimensions
Volume 230, Issue 1 (2023), pp. 1–92
Pub. online: 24 March 2023
Type: Article
Received
18 December 2019
18 December 2019
Accepted
23 May 2021
23 May 2021
Published
24 March 2023
24 March 2023
Abstract
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.