Proceedings of the American Mathematical Society

Author(s): Hua-Huai Chern; Jong-Shenq Guo; Chu-Pin Lo
Journal: Proc. Amer. Math. Soc. 131 (2003), 3191-3201.
MSC (2000): Primary 35B60, 34A12, 35B35
Posted: April 30, 2003
Retrieve article in: PDF

Abstract: We study a two-point free boundary problem for the curvature flow equation. By studying the corresponding nonlinear initial value problem, we obtain the existence and uniqueness of the forward self-similar solution of this problem. The corresponding curve is called the self-similar expanding curve. We also derive the asymptotic stability of this curve.

1.: S. J. Altschuler and L. F. Wu, Convergence to translating solutions for a class of quasilinear parabolic boundary problems, Math. Ann. 295 (1993), 761-765. MR 94d:35082
2.: S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. 2 (1994), 101-111. MR 97b:58032
3.: J. D. Buckmaster and G. S. S. Ludford, Theory of Laminar Flames, Cambridge University Press, Cambride, 1982. MR 84f:80011
4.: W. K. Burton, N. Cabrera, and F. C. Frank, The growth of crystals and equilibrium structure of their surfaces, Philos. Trans. Roy. Soc. London A 243 (1951), 299-358. MR 13:196f
5.: L. A. Caffarelli and J. L. Vazquez, A free boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc. 347 (1995), 411-441. MR 95e:35097
6.: Y.-G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geometry 33 (1991), 749-786. MR 93a:35093
7.: B. Chow and D. H. Tsai, Geometric expansion of convex plane curves, J. Diff. Geom. 44 (1996), 312-330. MR 97m:58041
8.: K. Deckelnick, C. M. Elliott, and G. Richardson, Long time asymptotics for forced curvature flow with applications to the motion of a superconducting vortex, Nonlinearity 10 (1997), 665-678. MR 98b:35080
9.: L. C. Evans and J. Spruck, Motion of level sets by mean curvature 1, J. Diff. Geom. 33 (1991), 635-681. MR 92h:35097
10.: V. A. Galaktionov, J. Hulshof and J. L. Vazquez, Extinction and focusing behaviour of spherical and annular flames described by a free boundary problem, J. Math. Pures Appl. 76 (1997), 563-608. MR 98h:35238
11.: Y. Giga, N. Ishimura, and Y. Kohsaka, Spiral solutions for a weakly anisotropic curvature flow equation, Hokkaido University Preprint Series in Mathematics, Series #529, June 2001.
12.: J.-S. Guo and Y. Kohsaka, Two-point free boundary problem for heat equation, preprint.
13.: M. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Clarendon Press, Oxford, UK, 1993. MR 97k:73001
14.: D. Hilhorst and J. Hulshof, A free boundary focusing problem, Proc. Amer. Math. Soc. 121 (1994), 1193-1202. MR 94j:35200
15.: G. Huisken, Non-parametric mean curvature evolution with boundary conditions, J. Diff. Equations 77 (1989), 369-378. MR 90g:35050
16.: H. Imai, N. Ishimura, and T. Ushijima, A crystalline motion of spiral-shaped curves with symmetry, J. Math. Anal. Appl. 240 (1999), 115-127. MR 2000j:53091
17.: J. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, New York, 1998. MR 2000c:92010
18.: Y. Kohsaka, Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary, Nonlinear Analysis 45 (2001), 865-894. MR 2002j:35320
19.: K. Mikula and D. Sevcovic, Evolution of plane curves driven by a nonlinear function of curvatures and anisotropy, SIAM J. Appl. Math. 61 (2001), 1473-1501. MR 2002b:65181
20.: H. Ninomiya and M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force, Free Boundary Problems: Theory and Applications I, Mathematical Sciences and Applications 13, Gakuto International Series, 2000, pp. 206-221. MR 2001j:53093
21.: H. Ninomiya and M. Taniguchi, Stability of traveling curved fronts in a curvature flow with driving force, Methods and Applications of Analysis 8 (2001), 429-450. MR 2003c:35012
22.: M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984. MR 86f:35034
23.: J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge, UK, 1999. MR 2000c:65015
24.: J. L. Vazquez, The free boundary problem for the heat equation with fixed gradient condition, Free boundary problems, theory and applications, Zakopane, Poland, Pitman Res. Notes in Math. Series 363, 1995, pp. 277-302. MR 98h:35246

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35B60, 34A12, 35B35

Hua-Huai Chern
Affiliation: Department of Computer and Information Sciences, National Taiwan Ocean University, 2, Pei-Ning Road, Keelung, Taiwan
Email: felix@cs.ntou.edu.tw

Jong-Shenq Guo
Affiliation: Department of Mathematics, National Taiwan Normal University, 88, S-4 Ting Chou Road, Taipei 117, Taiwan
Email: jsguo@math.ntnu.edu.tw

Chu-Pin Lo
Affiliation: Department of Applied Mathematics, Providence University, 200, Chung-Chi Road, Shalu, Taichung County 433, Taiwan
Email: cplo@pu.edu.tw

DOI: 10.1090/S0002-9939-03-07055-2
PII: S 0002-9939(03)07055-2
Received by editor(s): May 16, 2002
Posted: April 30, 2003
Communicated by: David S. Tartakoff
Copyright of article: Copyright 2003, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS