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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{cases}u_t ={\Delta}u+|u|^\frac{4}{n-2}u\;\;\;\text{in}\;\mathbb{R}^n\times(0,T)\\u(\centerdot,0)=u_0\;\;\;\text{in}\;\mathbb{R}^n\end{cases}$$\end{document} in dimension n = 5. More precisely we find that for given points q1,q2,...,qk and any sufficiently small T > 0 there is an initial condition u0 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3}{4}$$\end{document}. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles.