Type II blow-up in the 5-dimensional energy critical heat equation

We consider the Cauchy problem for the energy critical heat equation $$u_t=\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{\frac{4}{n-2}}u\text{in }{\mathbb{R}}^n\times (0,T),u(\cdot ,0)=u_0\text{in }{\mathbb{R}}^n$$ in dimension $n=5$. More precisely we find that for given points $q_1,q_2,\dots ,q_k$ and any sufficiently small $T>0$ there is an initial condition $u_0$ such that the solution $u(x,t)$ of the problem blows-up at exactly those $k$ points with rates type II, namely with absolute size $\sim (T-t{)}^{-\alpha }$ for $\alpha > \frac 34 $. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.