# Nonexistence of Type II Blowups for Energy-Critical heat Equation in Large Dimensions

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In this talk I will consider energy-critical nonlinear heat equation

$$ u_t=\Delta u+ u^{\frac{n+2}{n-2}}, u\geq 0 $$

We prove that for $n\geq 7$, any blow-up must be of Type I, i.e. the blow-up rate must be bounded by $(T-t)^{-\frac{n-2}{4}}$. The proof is built on several key ingredients: first we perform tangent ow analysis and study bubbling formation in this process; next we give a second order bubbling analysis in the multiplicity one case, where we use a reverse inner-outer gluing mechanism; finally, in the higher multiplicity case (bubbling tower/cluster), we develop Schoen's Harnack inequality and obtain next order estimates in Pohozaev identities for critical parabolic flows. (Joint work with Kelei Wang.)