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

TL;DR

This paper proves that for the energy critical nonlinear heat equation in dimensions n≥7 with nonnegative initial data, all blowups are of Type I, ruling out Type II blowups through advanced analytical techniques.

Contribution

It establishes the nonexistence of Type II blowups for high-dimensional energy critical heat equations with nonnegative initial data, using refined blowup analysis and gluing methods.

Findings

All blowups are of Type I for n≥7 with nonnegative initial data.

The proof employs a reverse inner-outer gluing mechanism.

The analysis includes detailed bubbling behavior and bubbling tower structures.

Abstract

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 the first blow up time. We prove that if $ n \geq 7$ and $ u_0 \geq 0$, then any blowup must be of Type I, i.e., \[\|u(\cdot, t)\|_{L^\infty({\mathbb R}^n)}\leq C(T-t)^{-\frac{1}{p-1}}.\] A similar result holds for bounded convex domains. The proof relies on a reverse inner-outer gluing mechanism and delicate analysis of bubbling behavior (bubbling tower/cluster).