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− u−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 topology u(t, r) − 1 λ Q r λ(t) →u∗ in H˙1 where the blow-up profile is given by the Talenti–Aubin soliton Q(r) = 1 1 + r2 8 , and with speed λ(t) ∼ T − t |log(T −t)|2 as 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]). © 2012 Elsevier Inc. All rights reserved.