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

We consider the problem v(t) = Delta(v) + vertical bar v vertical bar(p-1)v in Omega x (0, T), v = 0 on partial derivative Omega x (0, T), v > 0 in Omega x (0, T). In a domain Omega subset of R-d, d >= 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) = {infinity, if 3 <= d <= 10, 1 + 4/d-4-2 root d-1, if d >= 11. No type II radial blow-up is present for p < p(JL)(d). We take p = 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. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

In a domain with special symmetries of dimension d >= 7, a new solution with type II blow-up phenomenon for a power p less than the Joseph-Lundgren exponent is found. The solution blows up on the boundary in a negatively curved part in the form of a sharply scaled bubble, presenting a completely new phenomenon in a diffusion setting.