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

TL;DR

This paper constructs a novel finite-time blow-up solution for the energy supercritical heat equation in dimensions 5 to 7, where the singularity occurs along a shrinking sphere with a specific blow-up rate, revealing a new parabolic phenomenon.

Contribution

It introduces the first known type II blow-up solution with singularity along a shrinking sphere for the supercritical heat equation in certain dimensions.

Findings

Blow-up occurs along an (n-4)-dimensional shrinking sphere.

The solution exhibits a specific blow-up rate involving a logarithmic correction.

The singularity location approaches a fixed point as time approaches blow-up.

Abstract

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 $5\leq n\leq 7$, $\Omega=\R^n$ 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 $$u(x,t)\approx \la^{-1}(t)\frac{2\sqrt{2}}{1+\left|\frac{(r,z)-(\xi_r(t),\xi_z(t))}{\la(t)}\right|^2}$$ 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 new phenomenon in the parabolic setting.