Geometry driven Type II higher dimensional blow-up for the critical heat equation

TL;DR

This paper constructs the first example of type II blow-up solutions for the critical heat equation in higher dimensions with special symmetries, revealing a new blow-up phenomenon involving Aubin-Talenti bubbles on curved boundaries.

Contribution

It introduces a novel type II blow-up solution in higher dimensions for the critical heat equation, using geometric symmetry and boundary curvature, which was previously unknown.

Findings

First example of type II blow-up in this setting

Blow-up occurs on a boundary curve with negative curvature

Solution approaches a Dirac measure in energy density

Abstract

We consider the problem v_t & = \Delta v+ |v|^{p-1}v \quad\hbox{in }\ \Omega\times (0, T), v & =0 \quad\hbox{on } \partial \Omega\times (0, T ) , v& >0 \quad\hbox{in }\ \Omega\times (0, T) . In a domain $\Omega\subset \mathbb R^d$, $d\ge 7$ enjoying special symmetries, we find the first example of a solution with type II blow-up for a power $p$ less than the Joseph-Lundgren exponent $$p_{JL}(d)=\infty, & \text{if $3\le d\le 10$}, 1+{4\over d-4-2\,\sqrt{d-1}}, & \text{if $d\ge11$}. $$ No type II radial blow-up is present for $p< p_{JL}(d)$. We take $p=\frac{d+1}{d-3}$, the Sobolev critical exponent in one dimension less. The solution blows up on circle contained in a negatively curved part of the boundary in the form of a sharply scaled Aubin-Talenti bubble, approaching its energy density a Dirac measure for the curve. This is a completely new phenomenon for a diffusion setting.