Vol. 13, No. 1, 2020

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

Manuel del Pino, Monica Musso and Juncheng Wei

Vol. 13 (2020), No. 1, 215–274
Abstract

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

ut = Δu + u5 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) tγ1 2  if 1 < γ < 2,u(,t)t if γ > 2,u(,t)t(lnt)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).

Keywords
blow-up, critical exponents, nonlinear parabolic equations
Mathematical Subject Classification 2010
Primary: 35B33, 35B40, 35K58
Milestones
Received: 22 April 2018
Accepted: 29 December 2018
Published: 6 January 2020
Authors
Manuel del Pino
Department of Mathematical Sciences
University of Bath
Bath
United Kingdom
Monica Musso
Department of Mathematical Sciences
University of Bath
Bath
United Kingdom
Juncheng Wei
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada