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

Abstract

We consider the energy supercritical heat equation with the $(n-3)$-th Sobolev exponent \begin{equation*} \begin{cases} u_t=Δu+u^{3},~&\mbox{ in } Ω\times (0,T),\\ u(x,t)=u|_{\partialΩ},~&\mbox{ on } \partialΩ\times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } Ω, \end{cases} \end{equation*} where $5\le n\le 7$, $\Omega ={\mathbb{R}}^n$ or $\Omega \subset {\mathbb{R}}^n$ 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 $\Omega$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)\approx \la^{-1}(t)\frac{2\sqrt{2}}{1+\left|\frac{(r,z)-(ξ_r(t),ξ_z(t))}{\la(t)}\right|^2}$$ 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 $$(ξ_r(t),ξ_z(t))\sim (\sqrt{2(n-4)(T-t)},z_0)~\mbox{ as }~t\nearrow T,$$ 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.

arXiv comment64 pages; comments are welcome

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Cited by in corpus (1)

• Refined blowup analysis and nonexistence of Type II blowups for an energy critical nonlinear heat equation