A type II blowup for the six dimensional energy critical heat equation

We study blowup solutions of the 6D energy critical heat equation $u_t=\Delta u+|u|^{p-1}u$ in $\R^n\times(0,T)$. A goal of this paper is to show the existence of type II blowup solutions predicted by Filippas, Herrero and Vel\'azquez \cite{FilippasHV}. The dimension six is a border case whether a type II blowup can occur or not. Therefore the behavior of the solution is quite different from other cases. In fact, our solution behaves like $$u(x,t)\approx \begin{cases} \lambda(t)^{-2}{\sf Q}(\lambda(t)^{-1}x) & \text{in the inner region: } |x|\sim\lambda(t), -(p-1)^\frac{1}{p-1}(T-t)^{-\frac{1}{p-1}} & \text{in the selfsimilar region: } |x|\sim\sqrt{T-t} \end{cases}$$ with $\lambda(t)=(1+o(1))(T-t)^\frac{5}{4}|\log(T-t)|^{-\frac{15}{8}}$. The local energy $E_\text{loc}(u) =\frac{1}{2}\|\nabla u\|_{L^2(|x|<1)}^2-\frac{1}{3}\|u\|_{L^3(|x|<1)}^3$ of the solution goes to $-\infty$.