Infinite time blow-up for the three dimensional energy critical heat equation in bounded domains

Giacomo Ageno, Manuel Del Pino

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We consider the Dirichlet problem for the energy-critical heat equation (Formula presented.) where Ω is a bounded smooth domain in R 3. Let H γ(x,y) be the regular part of the Green function of -Δ-γ in Ω, where γ∈(0,λ 1) and λ 1 is the first Dirichlet eigenvalue of -Δ. Then, given a point q∈Ω such that 3γ(q)<λ 1, where (Formula presented.) we prove the existence of a non-radial global positive and smooth solution u(x, t) which blows up in infinite time with spike in q. The solution has the asymptotic profile (Formula presented.) where (Formula presented.)

Original languageEnglish
Number of pages95
JournalMathematische Annalen
Early online date15 May 2024
Publication statusE-pub ahead of print - 15 May 2024


The authors express gratitude to the anonymous referee for their careful reading, which contributed to the improvement of the manuscript.

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