Infinite time blow-up for the fractional heat equation with critical exponent

Monica Musso, Y. Sire, Juncheng Wei, Youquan Zheng, Yifu Zhou

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

We consider positive solutions for the fractional heat equation with critical exponent {ut=-(-Δ)su+un+2sn-2sinΩ×(0,∞),u=0on(Rn\Ω)×(0,∞),u(·,0)=u0inRn,where Ω is a smooth bounded domain in Rn, n> 4 s, s∈ (0 , 1) , u: Rn× [0 , ∞) → R and u0 is a positive smooth initial datum with u0|Rn\Ω=0. We prove the existence of u0 such that the solution blows up precisely at prescribed distinct points q1, … , qk in Ω as t→ + ∞. The main ingredient of the proofs is a new inner–outer gluing scheme for the fractional parabolic problems. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
Original languageEnglish
JournalMathematische Annalen
DOIs
Publication statusE-pub ahead of print - 1 Dec 2018

Cite this

Infinite time blow-up for the fractional heat equation with critical exponent. / Musso, Monica; Sire, Y.; Wei, Juncheng; Zheng, Youquan; Zhou, Yifu.

In: Mathematische Annalen, 01.12.2018.

Research output: Contribution to journalArticle

Musso, Monica ; Sire, Y. ; Wei, Juncheng ; Zheng, Youquan ; Zhou, Yifu. / Infinite time blow-up for the fractional heat equation with critical exponent. In: Mathematische Annalen. 2018.
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AB - We consider positive solutions for the fractional heat equation with critical exponent {ut=-(-Δ)su+un+2sn-2sinΩ×(0,∞),u=0on(Rn\Ω)×(0,∞),u(·,0)=u0inRn,where Ω is a smooth bounded domain in Rn, n> 4 s, s∈ (0 , 1) , u: Rn× [0 , ∞) → R and u0 is a positive smooth initial datum with u0|Rn\Ω=0. We prove the existence of u0 such that the solution blows up precisely at prescribed distinct points q1, … , qk in Ω as t→ + ∞. The main ingredient of the proofs is a new inner–outer gluing scheme for the fractional parabolic problems. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

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