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ON THE STABILITY OF STATIONARY LINE AND GRIM REAPER IN PLANAR CURVATURE FLOW

Part of: Global differential geometry Parabolic equations and systems

Published online by Cambridge University Press:  07 February 2011

XIAOLIU WANG  and
WEIFENG WO
XIAOLIU WANG*
Affiliation:
Department of Mathematics, Southeast University, No. 2 Sipailou, Nanjing 210096, PR China (email: [email protected])
WEIFENG WO
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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The asymptotic stability of two types of invariant solutions under a curvature flow in the whole plane is studied. First, by extending the work of others, we prove that the stationary line with nonzero slope will attract the graphical curves which surround it. Then a similar property is obtained for the grim reaper.

Keywords

curvature flowasymptotic behaviorstability

MSC classification

Secondary: 35K55: Nonlinear parabolic equations 35K15: Initial value problems for second-order parabolic equations 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 177 - 188
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This work is supported by the Grant for New PhD Teacher Program of Southeast University.

References

[1]Angenent, S., ‘The zero set of a solution of a parabolic equation’, J. reine angew. Math. 390 (1988), 79–96.Google Scholar
[2]Angenent, S., ‘Parabolic equations for curves on surfaces II. Intersections, blow up and generalized solutions’, Ann. of Math. (2) 133 (1991), 171–215.CrossRefGoogle Scholar
[3]Bakas, I. and Sourdis, C., ‘Dirichlet sigma models and mean curvature flow’, J. High Energy Phys. 6 (2007), 057.CrossRefGoogle Scholar
[4]Chou, K. S. and Zhu, X. P., ‘Shortening complete plane curves’, J. Differential Geom. 50 (1998), 471–504.CrossRefGoogle Scholar
[5]Chou, K. S. and Zhu, X. P., The Curve Shortening Problem (Chapman & Hall/CRC, Boca Raton, FL, 2001).CrossRefGoogle Scholar
[6]Ecker, K. and Huisken, G., ‘Mean curvature evolution of entire graphs’, Ann. of Math. (2) 130 (1989), 453–471.CrossRefGoogle Scholar
[7]Gage, M. and Hamilton, R., ‘The heat equation shrinking convex plane curves’, J. Differential Geom. 23 (1986), 69–96.CrossRefGoogle Scholar
[8]Grayson, M. A., ‘The heat equation shrinks embedded plane curves to round points’, J. Differential Geom. 26 (1987), 285–314.CrossRefGoogle Scholar
[9]Huisken, G., ‘A distance comparison principle for evolving curves’, Asian J. Math. 2 (1998), 127–133.CrossRefGoogle Scholar
[10]Ishimura, N., ‘Curvature evolution of plane curves with prescribed opening angle’, Bull. Aust. Math. Soc. 52 (1995), 287–296.CrossRefGoogle Scholar
[11]Lieberman, G. M., Second Order Parabolic Differential Equations (World Scientific, River Edge, NJ, 1996).CrossRefGoogle Scholar
[12]Mullins, W. W., ‘Two-dimensional motion of idealized grain boundaries’, J. Appl. Phys. 27 (1956), 900–904.CrossRefGoogle Scholar
[13]Nara, M. and Taniguchi, M., ‘The condition on the stability of stationary lines in a curvature flow in the whole plane’, J. Differential Equations 237 (2007), 61–76.CrossRefGoogle Scholar
[14]Polden, A., ‘Evolving curves’, Honours Thesis, Australian National University, 1991.Google Scholar
[15]Sapiro, G., Geometric Partial Differential Equations and Image Analysis (Cambridge University Press, Cambridge, 2001).CrossRefGoogle Scholar
[16]Stavrou, N., ‘Selfsimilar solutions to the mean curvature flow’, J. reine angew. Math. 499 (1998), 189–198.CrossRefGoogle Scholar
[17]Visintin, A., Models of Phase Transitions, Progress in Nonlinear Partial Differential Equations and Their Applications, 28 (Birkhäuser, Boston, 1996).CrossRefGoogle Scholar