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 partial derivative(t)u - Delta u - u(3) = 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/lambda Q(r/lambda(t)) -> u* in (H) over dot(vertical bar) where the blow-up profile is given by the Talenti-Aubin soliton Q(r) = 1/1+r(2)/8, and with speed lambda(t) similar to T - t/vertical bar log(T - t)vertical bar(2) as t -> T. Our approach uses a robust energy method approach developed for the study of geometrical dispersive problems (Raphael and Rodnianski, 2012 [18], Merle et al., 2011 [15]), and lies in the continuation of the study of the energy critical ham-ionic heat flow (Raphael and Schweyer, 2011 [19]) and the energy critical four dimensional wave equation (Hillaret and Raphael, 2010 [5]). (C) 2012 Elsevier Inc. All rights reserved.