Expanding convex immersed closed plane curves Authors Tai-Chia Lin Department of Mathematics National Taiwan University Chi-Cheung Poon Department of Mathematics National Chung Cheng University Dong-Ho Tsai Department of Mathematics National Tsing Hua University Article

First Online: 29 April 2008 Received: 17 January 2007 Accepted: 30 March 2008 DOI :
10.1007/s00526-008-0180-7

Cite this article as: Lin, T., Poon, C. & Tsai, D. Calc. Var. (2009) 34: 153. doi:10.1007/s00526-008-0180-7
Abstract We study the evolution driven by curvature of a given convex immersed closed plane curve. We show that it will converge to a self-similar solution eventually. This self-similar solution may or may not contain singularities. In case it does, we also have estimate on the curvature blow-up rate.

Mathematics Subject Classification (2000) 35K15 35K55 Download to read the full article text

References 1.

Andrews, B.: Evolving convex curves. Cal. Var. PDEs.

7 (4), 315–371 (1998)

MATH CrossRef 2.

Andrews, B.: Classification of limiting shapes for isotropic curve flows. J. AMS

16 (2), 443–459 (2003)

MATH 3.

Angenent, S.: The zero set of a solution of a parabolic equation. J. die Reine Angewandte Math.

390 , 79–96 (1988)

MATH MathSciNet CrossRef 4.

Angenent, S.: On the formation of singularities in the curve shortening flow. J. Differ. Geom.

33 , 601–633 (1991)

MATH MathSciNet 5.

Chen, X.Y., Matano, H.: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations. J. Differ. Equ.

78 (1), 160–190 (1989)

MATH CrossRef MathSciNet 6.

Cortazar, C., Del Pino, M., Elgueta, M.: On the blow-up set for

\({\partial_{t}u = \bigtriangleup u^{m} + u^{m}, m > 1, }\) . Indiana Univ. Math. J.

47 , 541–561 (1998)

MATH CrossRef MathSciNet 7.

Chow, B., Tsai, D.H.: Geometric expansion of convex plane curves. J. Differ. Geom.

44 , 312–330 (1996)

MATH MathSciNet 8.

Chou, K.S., Zhu, X.-P.: The Curve Shortening Problem. Chapman and Hall/CRC, London (2000)

9.

Feireisl, E., Simondon, F.: Convergence for degenerate parabolic equations. J. Differ. Equ.

152 , 439–466 (1999)

MATH CrossRef MathSciNet 10.

Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom.

23 , 69–96 (1986)

MATH MathSciNet 11.

Lin, Y.-C., Poon, C.C., Tsai, D.H.: Contracting convex immersed closed plane curves with fast speed (in preparation) (2008)

12.

Lin, Y.-C., Poon, C.C., Tsai, D.H.: Contracting convex immersed closed plane curves with slow speed (in preparation) (2008)

13.

Matano, H.: Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ.

18 (2), 221–227 (1978)

MATH MathSciNet 14.

Tsai, D.H.: Blowup and convergence of expanding immersed convex plane curves. Comm. Anal. Geom.

8 (4), 761–794 (2000)

MATH MathSciNet 15.

Tsai, D.H.: Behavior of the gradient for solutions of parabolic equations on the circle. Cal. Var. PDEs.

23 , 251–270 (2005)

MATH CrossRef 16.

Tsai, D.H.: Blowup behavior of an equation arising from plane curves expansion. Differ. Integ. Eq.

17 (7–8), 849–872 (2004)

MATH 17.

Urbas, J.: An expansion of convex hypersurfaces. J. Differ. Geom.

33 , 91–125 (1991)

MATH MathSciNet 18.

Urbas, J.: Convex curves moving homothetically by negative powers of their curvature. Asian J. Math.

3 (3), 635–658 (1999)

MATH MathSciNet