 
Summary: SIAM J. MATH. ANAL. c 2009 Society for Industrial and Applied Mathematics
Vol. 41, No. 3, pp. 11211137
THE RADIUS OF VANISHING BUBBLES IN EQUIVARIANT
HARMONIC MAP FLOW FROM D2
TO S2
S. B. ANGENENT, J. HULSHOF, AND H. MATANO§
Abstract. We derive an upper bound for the radius R(t) of a vanishing bubble in a family of
equivariant maps Ft : D2 S2 which evolve by the harmonic map flow. The selfsimilar "type 1"
radius would be R(t) = C
T  t. We prove that R(t) = o(T  t).
Key words. harmonic map flow, asymptotics, singularities
AMS subject classifications. 35K55, 53C44
DOI. 10.1137/070706732
1. Introduction. Let Nn
Rk
be a smooth submanifold. The Dirichlet integral
or energy of a map F from the unit disc D2
R2
into N is defined to be
