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
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 self-similar "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
1. Introduction. Let Nn
be a smooth submanifold. The Dirichlet integral
or energy of a map F from the unit disc D2
into N is defined to be