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

Abstract

We consider the problem v_t & = Δv+ |v|^{p-1}v \quad\hbox{in }\ Ω\times (0, T), v & =0 \quad\hbox{on } \partial Ω\times (0, T ) , v& >0 \quad\hbox{in }\ Ω\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.

49 pages, no figures. arXiv admin note: text overlap with arXiv:1705.01672

References in corpus (1)

• On strongly anisotropic type II blow up

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