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

We consider the Cauchy problem for the energy critical heat equation {ut=Δu+|u|4n−2uinRn×(0,T)u(⋅,0)=u0inRn 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) ^-α for α > 34. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles.