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The radius of vanishing bubbles in equivariant harmonic map flow from \(D^2\) to \(S^2\). (English) Zbl 1223.35198

In this paper, the authors construct a family of equivariant maps \(F_t:D^2\to S^2\), \(0\leq t<T\), that evolve by the harmonic map flow with a singularity at the origin at time \(T\) such that the radius \(R(t)\) of the vanishing bubble satisfies \(R(t)=o(T-t)\) as \(t\nearrow T\). This estimate differs from the formal asymptotics obtained by J. B. van den Berg, J. Hulshof and J. R. King [SIAM J. Appl. Math. 63, No. 5, 1682–1717 (2003; Zbl 1037.35023)] by a logarithmic factor \(\log (T-t)\) and from the estimate of P. Topping [Math. Z. 247, No. 2, 279–302 (2004; Zbl 1067.53055)] by a factor \((T-t)^{1/2+o(1)}\).

MSC:

35K55 Nonlinear parabolic equations
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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