Type II blow-up for the four dimensional energy critical semi linear heat equation

We consider the energy critical four dimensional semi linear heat equation ∂tu−Δu−u3=0∂tu−Δu−u3=0. We show the existence of type II finite time blow-up solutions and give a sharp description of the corresponding singularity formation. These solutions concentrate a universal bubble of energy in the critical topologyu(t,r)−1λQ(rλ(t))→u⁎in H˙1 where the blow-up profile is given by the Talenti–Aubin solitonQ(r)=11+r28, and with speedλ(t)∼T−t|log(T−t)|2as t→T. Our approach uses a robust energy method approach developed for the study of geometrical dispersive problems (Raphaël and Rodnianski, 2012 [18], Merle et al., 2011 [15]), and lies in the continuation of the study of the energy critical harmonic heat flow (Raphaël and Schweyer, 2011 [19]) and the energy critical four dimensional wave equation (Hillaret and Raphaël, 2010 [5]).