Type II Blow-up in the 5-dimensional Energy Critical Heat Equation

We consider the Cauchy problem for the energy critical heat equation in dimension n = 5. More precisely we find that for given points q(1),q(2),...,q(k) and any sufficiently small T > 0 there is an initial condition u(0) such that the solution u(x,t) of (0.1) blows-up at exactly those k points with rates type II, namely with absolute size T-t)(-) for . The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.