×

Asymptotic shape of cusp singularities in curve shortening. (English) Zbl 0829.35058

The paper deals with the smooth solution \(\gamma (t)\), \(t \in (0,T)\) of curve shortening which becomes singular at time \(t = T\). To this end the authors find the asymptotics for singular solutions of the equation \(k_t = k^2 (k_{\theta \theta} + k)\), where \(k (\theta + 2m \pi, t) \equiv k (\theta, t)\), \(k (\theta,t)\) is a curvature of \(\gamma\).
Reviewer: E.Minchev (Sofia)

MSC:

35K65 Degenerate parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. J. Altschuler, Singularities of the curve shrinking flow for space curves , J. Differential Geom. 34 (1991), no. 2, 491-514. · Zbl 0754.53006
[2] S. B. Angenent, On the formation of singularities in the Curve Shortening Flow , J. Differential Geom. 33 (1991), no. 3, 601-633. · Zbl 0731.53002
[3] S. Angenent, The zeroset of a solution of a parabolic equation , J. Reine Angew. Math. 390 (1988), 79-96. · Zbl 0644.35050 · doi:10.1515/crll.1988.390.79
[4] S. D. Eidelman, Parabolic Systems , Translated from the Russian by Scripta Technica, London, North-Holland, Amsterdam, 1969. · Zbl 0181.37403
[5] A. Friedman and B. McLeod, Blow-up of solutions of nonlinear degenerate parabolic equations , Arch. Rational Mech. Anal. 96 (1986), no. 1, 55-80. · Zbl 0619.35060
[6] M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves , J. Differential Geom. 23 (1986), no. 1, 69-96. · Zbl 0621.53001
[7] M. Grayson, The heat equation shrinks embedded plane curves to round points , J. Differential Geom. 26 (1987), no. 2, 285-314. · Zbl 0667.53001
[8] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation , J. Fac. Sci. Univ. Tokyo, Sect. IA 29 (1982), no. 2, 401-441. · Zbl 0496.35011
[9] J. J. L. Velázquez, Blow-up for semilinear parabolic equations , to appear in Recent Advances in Partial Differential Equations, ed. by M. A. Herrero and E. Zuazua, Masson, 1994.
[10] P. A. Watterson, Force-free magnetic evolution in the reversed-field pinch , Thesis, Cambridge University, 1985.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.