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

Abstract We consider the problem v t = Δ v + | v | p − 1 v in  Ω × ( 0 , T ) , v = 0 on  ∂ Ω × ( 0 , T ) , v > 0 in  Ω × ( 0 , T ) . In a domain Ω ⊂ 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 J L ( d ) = { ∞ , if  3 ≤ d ≤ 10 , 1 + 4 d − 4 − 2 d − 1 , if  d ≥ 11 . No type II radial blow-up is present for p p J L ( 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.