Nonexistence of pure bubble type II blow-up solutions to energy critical wave equation in the 3D radial case

TL;DR

This paper proves that in the 3D radial focusing energy-critical wave equation, pure bubble type II blow-up solutions cannot exist, confirming that the free wave component is always present in the soliton resolution.

Contribution

It establishes the nonexistence of pure bubble type II blow-up solutions in the radial 3D energy-critical wave equation, refining the understanding of blow-up behavior.

Findings

Pure bubble type II solutions do not exist in the radial 3D case.

The free wave component is always non-zero in the soliton resolution.

Supports the soliton resolution conjecture in the radial setting.

Abstract

In this work we consider the focusing, energy-critical wave equation in 3D radial case. According to the soliton resolution conjecture, which has been verified in the radial case, any 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 the blow-up time. We prove that there does not exist any pure bubble type II blow up solutions. In other words, the free wave part is never zero in the soliton resolution of type II blow-up solutions, regardless of the bubble number.