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

TL;DR

This paper constructs a specific type II blowup solution for the six-dimensional energy critical heat equation, demonstrating a new blowup behavior at the critical dimension where such solutions are predicted but not previously confirmed.

Contribution

It provides the first explicit construction of a type II blowup solution in 6D, confirming theoretical predictions and analyzing its detailed asymptotic behavior.

Findings

Existence of type II blowup solutions in 6D confirmed.

Solution exhibits a specific asymptotic profile with a logarithmic correction.

Local energy of the solution diverges to negative infinity.

Abstract

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 $-\infty$.