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).

PDF Access Denied

We have not been able to recognize your IP address 18.226.28.197 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors