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We consider the energy supercritical heat equation with the $(n-3)$-thSobolev 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 $5\leq n\leq 7$, $\Omega=\R^n$ or $\Omega \subset \R^n$is a smooth, bounded domain enjoying special symmetries. We construct type IIfinite time blow-up solution $u(x,t)$ with the singularity taking place alongan $(n-4)$-dimensional {\em shrinking sphere} in $\Omega$. More precisely, atleading order, the solution $u(x,t)$ is of the sharply scaled form$$u(x,t)\approx\la^{-1}(t)\frac{2\sqrt{2}}{1+\left|\frac{(r,z)-(\xi_r(t),\xi_z(t))}{\la(t)}\right|^2}$$where $r=\sqrt{x_1^2+\cdots+x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with$x=(x_1,\cdots,x_n)\in\Omega$. Moreover, the singularity location$$(\xi_r(t),\xi_z(t))\sim (\sqrt{2(n-4)(T-t)},z_0)~\mbox{ as }~t\nearrow T,$$for some fixed $z_0$, and the blow-up rate $$\la(t)\sim\frac{T-t}{|\log(T-t)|^2}~\mbox{ as }~t\nearrow T.$$ This is a completely newphenomenon in the parabolic setting.

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