#### 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

For each $\gamma >1$ we find initial data (not necessarily radially symmetric) with $\underset{|x|\to \infty }{lim}|x{|}^{\gamma }{u}_{0}\left(x\right)>0$ such that as $t\to \infty$

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