Analysis & PDE

  • Anal. PDE
  • Volume 13, Number 1 (2020), 215-274.

Infinite-time blow-up for the 3-dimensional energy-critical heat equation

Manuel del Pino, Monica Musso, and Juncheng Wei

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We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension 3

u t = Δ u + u 5  in  3 × ( 0 , ) , u ( x , 0 ) = u 0 ( x )  in  3 .

For each γ>1 we find initial data (not necessarily radially symmetric) with lim|x||x|γu0(x)>0 such that as t

u ( , t ) t γ 1 2  if  1 < γ < 2 , u ( , t ) t  if  γ > 2 , u ( , t ) t ( ln t ) 1  if  γ = 2 .

Furthermore we show that this infinite-time blow-up is codimensional-1 stable. The existence of such solutions was conjectured by Fila and King (Netw. Heterog. Media 7:4 (2012), 661–671).

Article information

Anal. PDE, Volume 13, Number 1 (2020), 215-274.

Received: 22 April 2018
Accepted: 29 December 2018
First available in Project Euclid: 16 January 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 35B33: Critical exponents 35B40: Asymptotic behavior of solutions 35K58: Semilinear parabolic equations

blow-up critical exponents nonlinear parabolic equations


del Pino, Manuel; Musso, Monica; Wei, Juncheng. Infinite-time blow-up for the 3-dimensional energy-critical heat equation. Anal. PDE 13 (2020), no. 1, 215--274. doi:10.2140/apde.2020.13.215.

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