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We consider the energy critical semilinear heat equation $$\left\{\begin{aligned} &\partial_t u-\Delta u =|u|^{\frac{4}{n-2}}u &\mbox{in }{\mathbb R}^n\times(0,T),\\ &u(x,0)=u_0(x), \end{aligned}\right. $$ where $n\geq 3$, $u_0\in L^\infty({\mathbb R}^n)$, and $T\in {\mathbb R}^+$ is thefirst blow up time. We prove that if $ n \geq 7$ and $ u_0 \geq 0$, then anyblowup must be of Type I, i.e., \[\|u(\cdot, t)\|_{L^\infty({\mathbb R}^n)}\leqC(T-t)^{-\frac{1}{p-1}}.\] A similar result holds for bounded convex domains.The proof relies on a reverse inner-outer gluing mechanism and delicateanalysis of bubbling behavior (bubbling tower/cluster).

Refined blowup analysis and nonexistence of Type II blowups for an energy critical nonlinear heat equation