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 } Omegatimes (0,T), u(x,t)=u|_{partialOmega},~&mbox{ on } partialOmegatimes (0,T), u(x,0)=u_0(x),~&mbox{ in } Omega, end{cases} end{equation*} where $5leq nleq 7$, $Omega=R^n$ or $Omega subset R^n$ 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 $Omega$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)approx la^{-1}(t)frac{2sqrt{2}}{1+left|frac{(r,z)-(xi_r(t),xi_z(t))}{la(t)}right|^2}$$ where $r=sqrt{x_1^2+cdots+x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with $x=(x_1,cdots,x_n)inOmega$. Moreover, the singularity location $$(xi_r(t),xi_z(t))sim (sqrt{2(n-4)(T-t)},z_0)~mbox{ as }~t earrow T,$$ for some fixed $z_0$, and the blow-up rate $$la(t)sim frac{T-t}{|log(T-t)|^2}~mbox{ as }~t earrow T.$$ This is a completely new phenomenon in the parabolic setting.