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

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.