New type II Finite time blow-up for the energy supercritical heat equation

We consider the energy supercritical heat equation with the $(n-3)$-th Sobolev exponent \begin{equation*} \begin{cases} u_t=\Delta u+u^{3},~&\mbox{ in } \Omega\times (0,T),\ u(x,t)=u|_{\partial\Omega},~&\mbox{ on } \partial\Omega\times (0,T),\ u(x,0)=u_0(x),~&\mbox{ in } \Omega, \end{cases} \end{equation*} where $5\le n\le 7$, $\mathrm{\Omega }={\mathbb{R}}^n$ or $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a smooth, bounded domain enjoying special symmetries. We construct type II finite time blow-up solution $u(x,t)$ with the singularity taking place along an $(n-4)$-dimensional {\em shrinking sphere} in $\mathrm{\Omega }$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)\approx {\text{la}}^{-1}(t)\frac{2\sqrt{2}}{1+{\mid \frac{(r,z)-({\xi }_r(t),{\xi }_z(t))}{\text{la}(t)}\mid }^2}$$ where $r=\sqrt{x_1^2+\cdots +x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with $x=(x_1,\cdots ,x_n)\in \mathrm{\Omega }$. Moreover, the singularity location $$({\xi }_r(t),{\xi }_z(t))\sim (\sqrt{2(n-4)(T-t)},z_0)ast\nearrow T,$$ for some fixed $z_0$, and the blow-up rate $$\text{la}(t)\sim \frac{T-t}{\mathrm{\mid }\log (T-t){\mathrm{\mid }}^2}ast\nearrow T\mathrm{.}$$ This is a completely new phenomenon in the parabolic setting.