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

Abstract: We consider the Cauchy problem for the energy critical heat equation $$ u_t = \Delta u + |u|^{\frac 4{n-2}}u {{\quad\hbox{in } }} \ {\mathbb R}^n \times (0, T), \quad u(\cdot,0) =u_0 {{\quad\hbox{in } }} {\mathbb R}^n $$ in dimension $n=5$. More precisely we find that for given points $q_1, q_2,\ldots, q_k$ 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.