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 $$u(x,t)\approx \{\begin{array}{lll}\lambda (t{)}^{-2}\mathsf{Q}(\lambda (t{)}^{-1}x)& \text{in the inner region: }\mathrm{\mid }x\mathrm{\mid }\sim \lambda (t),-(p-1{)}^{\frac{1}{p-1}}(T-t{)}^{-\frac{1}{p-1}}& \text{in the selfsimilar region: }\mathrm{\mid }x\mathrm{\mid }\sim \sqrt{T-t}\end{array}$$ with $\lambda (t)=(1+o(1))(T-t{)}^{\frac{5}{4}}\mathrm{\mid }\log (T-t){\mathrm{\mid }}^{-\frac{15}{8}}$. 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 $-\mathrm{\infty }$.