A Type II Blowup for the Six Dimensional Energy Critical Heat Equation

Abstract

We study blowup solutions of the 6D energy critical heat equation \(u_t=\Delta u+|u|^{p-1}u\) in \({\mathbb {R}}^n\times (0,T)\). A goal of this paper is to show the existence of type II blowup solutions predicted by Filippas et al. (R Soc Lond Proc Ser A Math Phys Eng Sci 456(2004):2957–2982, 2000). 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

$$\begin{aligned} u(x,t)\approx {\left\{ \begin{array}{ll} \lambda (t)^{-2}{{\textsf {Q}}}(\lambda (t)^{-1}x) &{} {\text {in the inner region: }} |x|\sim \lambda (t),\\ -(p-1)^\frac{1}{p-1}(T-t)^{-\frac{1}{p-1}} &{} {\text {in the selfsimilar region: }} |x|\sim \sqrt{T-t} \end{array}\right. } \end{aligned}$$

with \(\lambda (t)=(1+o(1))(T-t)^\frac{5}{4}|\log (T-t)|^{-\frac{15}{8}}\). Particularly the local energy defined by \(E_{\text {loc}}(u(t)) =\frac{1}{2}\Vert \nabla u(t)\Vert _{L^2(|x|<1)}^2-\frac{1}{p+1}\Vert u(t)\Vert _{L^{p+1}(|x|<1)}^{p+1}\) goes to \(-\infty \).