Vol. 10, No. 1, 2017

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

Volume 15
Issue 6, 1375–1616
Issue 5, 1131–1373
Issue 4, 891–1130
Issue 3, 567–890
Issue 2, 273–566
Issue 1, 1–272

Volume 14, 8 issues

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.
Nonradial type II blow up for the energy-supercritical semilinear heat equation

Charles Collot

Vol. 10 (2017), No. 1, 127–252

We consider the semilinear heat equation in large dimension d 11

tu = Δu + |u|p1u,p = 2q + 1,q ,

on a smooth bounded domain Ω d with Dirichlet boundary condition. In the supercritical range p p(d) > 1 + 4 d2, we prove the existence of a countable family (u) of solutions blowing up at time T > 0 with type II blow up:

u(t)L C(T t)c

with blow-up speed c > 1 p1. The blow up is caused by the concentration of a profile Q which is a radially symmetric stationary solution:

u(x,t) 1 λ(t) 2 p1 Q(x x0 λ(t) ),λ C(un)(T t)c(p1) 2 ,

at some point x0 Ω. The result generalizes previous works on the existence of type II blow-up solutions, which only existed in the radial setting. The present proof uses robust nonlinear analysis tools instead, based on energy methods and modulation techniques. This is the first nonradial construction of a solution blowing up by concentration of a stationary state in the supercritical regime, and it provides a general strategy to prove similar results for dispersive equations or parabolic systems and to extend it to multiple blow ups.

PDF Access Denied

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:

blow up, heat, soliton, ground state, nonlinear, nonradial, supercritical
Mathematical Subject Classification 2010
Primary: 35B44
Secondary: 35K58, 35B20
Received: 9 April 2016
Accepted: 29 September 2016
Published: 30 January 2017
Charles Collot
Laboratoire J.A. Dieudonné
Université de Nice Sophia Antipolis
Parc Valrose
%, 28 Avenue Valrose
06108 Cedex 02 Nice