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

For each $γ > 1$ we find initial data (not necessarily radially symmetric) with $lim_{| x | \to \infty} | x |^{γ} u_{0} (x) > 0$ such that as $t \to \infty$

∥ u (\cdot, t) ∥_{\infty} \sim t^{γ - \frac{1}{2}} if 1 < γ < 2, ∥ u (\cdot, t) ∥_{\infty} \sim \sqrt{t} if γ > 2, ∥ u (\cdot, t) ∥_{\infty} \sim \sqrt{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).

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

Vol. 13, No. 1, 2020

Manuel del Pino, Monica Musso and Juncheng Wei

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors