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

2017/10/29 by Manuel del Pino, Monica Musso, del Pino, Manuel +3 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1710.11461

openalex publication_date 2017/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem v_t & = Δv+ |v|^p-1v \hboxin Ω× (0, T), v & =0 \hboxon ∂ Ω× (0, T ) , v& >0 \hboxin Ω× (0, T) . In a domain Ω⊂ \mathbb 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)=∞, amp; if 3≤ d≤ 10, 1+4\over d-4-2 √(d-1), amp; 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.

Citations

• Quasilinear Dirichlet problems driven by positive sources

Cited by

• On blow-up rate for the Hénon parabolic equation with Sobolev supercritical nonlinearity
• Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

Related

• Type II blow-up in the 5-dimensional energy critical heat equation
• Refined blowup analysis and nonexistence of Type II blowups for an energy critical nonlinear heat equation
• New type II Finite time blow-up for the energy supercritical heat equation
• A type II blowup for the six dimensional energy critical heat equation
• Collapsing-tube type II blow-up for the energy-supercritical heat equation