Contemporary Mathematics
Volume 367, 2005
Singularities at
oo in Equivariant Harmonic Map Flow
Sigurd Angenent and Joost Hulshof
1. Introduction
Many nonlinear parabolic equations in Geometry and Applied Mathematics
(mean curvature flow, Ricci flow, harmonic map flow, the Yang-Mills flow, reaction
diffusion equations such as
have solutions which become singular
either in finite or infinite time, meaning either that the evolving object (map, metric,
surface, or function) becomes unbounded, or that one of its derivatives becomes
unbounded. The analysis of the asymptotic behaviour of a solution of a nonlinear
parabolic equation just before it becomes singular is known to be a difficult problem.
The main general point of this note is that this analysis is considerably easier in
the case where the singularity occurs in infinite time. The reason for this is that
infinite time singularities are a "stable phenomenon" in the following sense. Given
an initial data whose solution becomes singular at t
oo, a slight modification
of this initial data will generally still produce a solution which becomes singular
at the same time (namely, t
oo). In contrast, if a solution becomes singular
at time
t =
T oo, then a small perturbation of the initial data will generally
still produce a solution which becomes singular in finite time, but, usually, at a
different time (e.g. simply replace the solution u(t) by the solution u(t +E).) This
instability makes that standard tools for constructing solutions to PDEs (such as
the contraction mapping principle or the method of sub and supersolutions) cannot
directly provide precise information about solutions near their singularities. To
make all this more specific we now consider the example of harmonic map flow
from the disc D
JR.2 to the sphere 8
C JR.3

A family of maps
Ft :
evolves according to the harmonic map flow if
Ftl Ft.
2000 Mathematics Subject Classification. Primary 53C44, 53C43, 58J35.
The work presented in this note was done while the first author was visiting the Universiteit
Leiden in 2000/2001. During this year he was supported by NWO through Sjoerd Verduijn Lunel's
NWO grant, NWO 600-61-410. He greatfully acknowledges the hospitality he received from his
hosts, Sjoerd Verduijn Lunel and Bert Peletier.
The second author was supported by the RTN network Fronts-Singularities, HPRN-CT-2002-
0027 4 as well as the CWI in Amsterdam.
2005 American Mathematical Society
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