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Contemporary Mathematics Singularities at t = in Equivariant Harmonic Map Flow
 

Summary: Contemporary Mathematics
Singularities at t = 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 ut = u + up
) 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 = , a slight modification
of this initial data will generally still produce a solution which becomes singular
at the same time (namely, t = ). In contrast, if a solution becomes singular
at time t = T < , then a small perturbation of the initial data will generally
still produce a solution which becomes singular in finite time, but, usually, at a

  

Source: Angenent, Sigurd - Department of Mathematics, University of Wisconsin at Madison

 

Collections: Mathematics