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, Herrero and Velázquez \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 \[ u(x,t)\approx \begin{cases} λ(t)^{-2}{\sf Q}(λ(t)^{-1}x) & \text{in the inner region: } |x|\simλ(t), -(p-1)^\frac{1}{p-1}(T-t)^{-\frac{1}{p-1}} & \text{in the selfsimilar region: } |x|\sim\sqrt{T-t} \end{cases} \] with $\lambda (t)=(1+o(1))(T-t{)}^{\frac{5}{4}}|\log (T-t){|}^{-\frac{15}{8}}$. The local energy $E_{\text{loc}}(u)=\frac{1}{2}\Vert \nabla u{\Vert }_{L^2(|x|<1)}^2-\frac{1}{3}\Vert u{\Vert }_{L^3(|x|<1)}^3$ of the solution goes to $-\infty$.

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