Vol. 12, No. 1, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 14
Issue 6, 1671–1976
Issue 5, 1333–1669
Issue 4, 985–1332
Issue 3, 667–984
Issue 2, 323–666
Issue 1, 1–322

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow

Tej-eddine Ghoul, Slim Ibrahim and Van Tien Nguyen

Vol. 12 (2019), No. 1, 113–187

We consider the energy-supercritical harmonic heat flow from d into the d-sphere Sd with d 7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one-dimensional semilinear heat equation

tu = r2u + (d 1) r ru (d 1) 2r2 sin(2u).

We construct for this equation a family of C solutions which blow up in finite time via concentration of the universal profile

u(r,t) Q( r λ(t)),

where Q is the stationary solution of the equation and the speed is given by the quantized rates

λ(t) cu(T t) γ , ,2 > γ = γ(d) (1,2].

The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski (Camb. J. Math. 3:4 (2015), 439–617) for the energy supercritical nonlinear Schrödinger equation and by Raphaël and Schweyer (Anal. PDE 7:8 (2014), 1713–1805) for the energy critical harmonic heat flow. Then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed-point theorem. Moreover, our constructed solutions are in fact (1)-codimension stable under perturbations of the initial data. As a consequence, the case = 1 corresponds to a stable type II blowup regime.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address 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:

harmonic heat flow, blowup, stability, differential geometry
Mathematical Subject Classification 2010
Primary: 35B40, 35K50
Secondary: 35K55, 35K57
Received: 23 September 2017
Accepted: 9 April 2018
Published: 2 August 2018
Tej-eddine Ghoul
Department of Mathematics
New York University in Abu Dhabi
Abu Dhabi
United Arab Emirates
Slim Ibrahim
Department of Mathematics and Statistics
University of Victoria
Victoria, BC
Department of Mathematics
New York University in Abu Dhabi
Abu Dhabi
United Arab Emirates
Van Tien Nguyen
Department of Mathematics
New York University Abu Dhabi
Abu Dhabi
United Arab Emirates