Nonexistence of multi-bubble radial solutions to the 3D energy critical wave equation

TL;DR

This paper proves that in the 3D energy-critical radial wave equation, solutions cannot decompose into multiple solitons, providing a complete classification of their long-term behavior.

Contribution

It establishes the nonexistence of multi-soliton blow-up solutions in the 3D radial case, extending the understanding of soliton resolution.

Findings

No multi-soliton blow-up solutions exist in the 3D radial case.

Complete classification of asymptotic behaviors of radial solutions.

Soliton resolution involves at most one soliton in this setting.

Abstract

In this work we consider the focusing, energy-critical wave equation in 3D radial case. It has been verified that any global or type II blow-up solution decomposes into a superposition of several decoupled grounds states, a free wave and a small error, as time tends to infinity or the blow-up time. This is usually called soliton resolution. However, all known examples of soliton resolution in the 3D radial case come with no more than one soliton. In this work we prove the nonexistence of any global or type II blow up solution with two or more solitons, thus give a complete classification of asymptotic behaviours of radial solutions.