2014, 34(5): 2069-2090. doi: 10.3934/dcds.2014.34.2069

A fast blow-up solution and degenerate pinching arising in an anisotropic crystalline motion

1. 

Mathematical Sciences, College of Systems Engineering and Science, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama 337-8570, Japan

2. 

Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kanagawa 214-8571, Japan

Received  January 2013 Revised  July 2013 Published  October 2013

The asymptotic behavior of solutions to an anisotropic crystalline motion is investigated. In this motion, a solution polygon changes the shape by a power of crystalline curvature in its normal direction and develops singularity in a finite time. At the final time, two types of singularity appear: one is a single point-extinction and the other is degenerate pinching. We will discuss the latter case of singularity and show the exact blow-up rate for a fast blow-up or a type II blow-up solution which arises in an equivalent blow-up problem.
Citation: Tetsuya Ishiwata, Shigetoshi Yazaki. A fast blow-up solution and degenerate pinching arising in an anisotropic crystalline motion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2069-2090. doi: 10.3934/dcds.2014.34.2069
References:
[1]

B. Andrews, Evolving convex curves,, Calc. Var. Partial Differential Equations, 7 (1998), 315. doi: 10.1007/s005260050111.

[2]

B. Andrews, Singularities in crystalline curvature flows,, Asian J. Math., 6 (2002), 101.

[3]

S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure, 2. Evolution of an isothermal interface,, Arch. Rational Mech. Anal., 108 (1989), 323. doi: 10.1007/BF01041068.

[4]

M. Beneš, M. Kimura and S. Yazaki, Second order numerical scheme for motion of polygonal curves with constant area speed,, Interfaces and Free Boundaries, 11 (2009), 515. doi: 10.4171/IFB/221.

[5]

T. Fukui and Y. Giga, Motion of A Graph by Nonsmooth Weighted Curvature,, World Congress of Nonlinear Analysis '92 (ed. Lakshmikantham, (1996), 47.

[6]

Y. Giga, Moving boundary equations with anisotropic curvature (Japanese),, Sūgaku, 52 (2000), 113.

[7]

M.-H. Giga and Y. Giga, Crystalline and level set flow - convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane, Free boundary problems: Theory and applications I (Chiba, 1999),, GAKUTO Internat. Ser. Math. Sci. Appli., 13 (2000), 64.

[8]

Y. Giga and P. Rybka, Facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term,, J. Differential Equations, 246 (2009), 2264. doi: 10.1016/j.jde.2009.01.009.

[9]

P. M. Girāo, Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature,, SIAM J. Numer. Anal., 32 (1995), 886. doi: 10.1137/0732041.

[10]

P. M. Girāo and R. V. Kohn, Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature,, Numer. Math., 67 (1994), 41. doi: 10.1007/s002110050017.

[11]

M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane,, Oxford Mathematical Monographs. The Clarendon Press, (1993).

[12]

C. Hirota, T. Ishiwata and S. Yazaki, Some results on singularities of solutions to an anisotropic crystalline curvature flow,, Mathematical approach to nonlinear phenomena: Modelling, 23 (2005), 119.

[13]

C. Hirota, T. Ishiwata and S. Yazaki, C. Note on the Asymptotic Behavior of Solutions to An Anisotropic Crystalline Curvature Flow,, Recent Advances on Elliptic and Parabolic Issues: Proceedings of the 2004 Swiss-Japanese Seminar, (2006), 6.

[14]

C. Hirota, T. Ishiwata and S. Yazaki, Numerical study and examples on singularities of solutions to anisotropic crystalline curvature flows of nonconvex polygonal curves,, Asymptotic analysis and singularitieselliptic and parabolic PDEs and related problems, 47 (2007), 543.

[15]

K. Ishii and H. M. Soner, Regularity and convergence of crystalline motion,, SIAM J. Math. Anal., 30 (1999), 19. doi: 10.1137/S0036141097317347.

[16]

T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena,, Japan Journal of Industrial and Applied Mathematics, 25 (2008), 233.

[17]

T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect,, Discrete Contin. Dyn. Syst., 4 (2011), 865. doi: 10.3934/dcdss.2011.4.865.

[18]

T. Ishiwata, Motion of Polygonal Curved Fronts by Crystalline Motion: V-Shaped Solutions and Eventual Monotonicity,, Discrete Contin. Dyn. Syst. 2011, (2011), 717.

[19]

T. Ishiwata and M. Tsutsumi, Semidiscretization in space of nonlinear degenerate parabolic equations with blow-up of the solution,, J. Comput. Math., 18 (2000), 571.

[20]

T. Ishiwata, T. K. Ushijima, H. Yagisita and S. Yazaki, Two examples of nonconvex self-similar solution curves for a crystalline curvature flow,, Proc. Japan Acad. Ser. A Math. Sci., 80 (2004), 151.

[21]

T. Ishiwata and S. Yazaki, On the blow-up rate for fast blow-up solutions arising in an anisotropic crystalline motion,, The Proceedings of the Sixth Japan-China Joint Seminar; a special issue of J. Comp. App. Math., 159 (2003), 55. doi: 10.1016/S0377-0427(03)00556-9.

[22]

M. Kimura, D. Tagami and S. Yazaki, Polygonal Hele-Shaw problem with surface tension,, Interfaces and Free Boundaries, 15 (2013), 77. doi: 10.4171/IFB/295.

[23]

R. Kobayashi and Y. Giga, On anisotropy and curvature effects for growing crystals,, Recent topics in mathematics moving toward science and engineering, 18 (2001), 207. doi: 10.1007/BF03168571.

[24]

N. Mizoguchi, Type-II blowup for a semilinear heat equation,, Adv. Differential Equations, 9 (2004), 1279.

[25]

N. Mizoguchi, Rate of type II blowup for a semilinear heat equation,, Math. Ann., 339 (2007), 839. doi: 10.1007/s00208-007-0133-z.

[26]

N. Mizoguchi, Blow-up rate of type II and the braid group theory,, Trans. Amer. Math. Soc., 363 (2011), 1419. doi: 10.1090/S0002-9947-2010-04784-1.

[27]

A. Stancu, Asymptotic behavior of solutions to a crystalline flow,, Hokkaido Math. J., 27 (1998), 303.

[28]

J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry,, Differential geometry, (1991), 321.

[29]

J. E. Taylor, J. W. Cahn and C. A. Handwerker, Geometric models of crystal growth,, Acta Metall. Mater., 40 (1992), 1443.

[30]

T. K. Ushijima and S. Yazaki, Convergence of a crystalline algorithm for the motion of a closed convex curve by a power of curvature $V=K^\alpha$,, SIAM J. Numer. Anal., 37 (2000), 500. doi: 10.1137/S0036142997330135.

[31]

T. K. Ushijima and S. Yazaki, Convergence of a crystalline approximation for an area-preserving motion,, J. Comp. App. Math., 166 (2004), 427. doi: 10.1016/j.cam.2003.08.041.

[32]

J. H. Wilkinson, The Algebraic Eigenvalue Problem,, Clarendon press, (1965).

[33]

T. Yamamoto, Sūchikaisekinyūmon (in Japanese),, Saiensu-sha (1976, (2003).

[34]

S. Yazaki, Asymptotic behavior of solutions to an expanding motion by a negative power of crystalline curvature,, Adv. Math. Sci. Appl., 12 (2002), 227.

[35]

S. Yazaki, Motion of nonadmissible convex polygons by crystalline curvature,, Publications of Research Institute for Mathematical Sciences, 43 (2007), 155.

[36]

S. Yazaki, An area-preserving crystalline curvature flow equation,, Topics in mathematical modeling, 4 (2008), 169.

show all references

References:
[1]

B. Andrews, Evolving convex curves,, Calc. Var. Partial Differential Equations, 7 (1998), 315. doi: 10.1007/s005260050111.

[2]

B. Andrews, Singularities in crystalline curvature flows,, Asian J. Math., 6 (2002), 101.

[3]

S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure, 2. Evolution of an isothermal interface,, Arch. Rational Mech. Anal., 108 (1989), 323. doi: 10.1007/BF01041068.

[4]

M. Beneš, M. Kimura and S. Yazaki, Second order numerical scheme for motion of polygonal curves with constant area speed,, Interfaces and Free Boundaries, 11 (2009), 515. doi: 10.4171/IFB/221.

[5]

T. Fukui and Y. Giga, Motion of A Graph by Nonsmooth Weighted Curvature,, World Congress of Nonlinear Analysis '92 (ed. Lakshmikantham, (1996), 47.

[6]

Y. Giga, Moving boundary equations with anisotropic curvature (Japanese),, Sūgaku, 52 (2000), 113.

[7]

M.-H. Giga and Y. Giga, Crystalline and level set flow - convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane, Free boundary problems: Theory and applications I (Chiba, 1999),, GAKUTO Internat. Ser. Math. Sci. Appli., 13 (2000), 64.

[8]

Y. Giga and P. Rybka, Facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term,, J. Differential Equations, 246 (2009), 2264. doi: 10.1016/j.jde.2009.01.009.

[9]

P. M. Girāo, Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature,, SIAM J. Numer. Anal., 32 (1995), 886. doi: 10.1137/0732041.

[10]

P. M. Girāo and R. V. Kohn, Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature,, Numer. Math., 67 (1994), 41. doi: 10.1007/s002110050017.

[11]

M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane,, Oxford Mathematical Monographs. The Clarendon Press, (1993).

[12]

C. Hirota, T. Ishiwata and S. Yazaki, Some results on singularities of solutions to an anisotropic crystalline curvature flow,, Mathematical approach to nonlinear phenomena: Modelling, 23 (2005), 119.

[13]

C. Hirota, T. Ishiwata and S. Yazaki, C. Note on the Asymptotic Behavior of Solutions to An Anisotropic Crystalline Curvature Flow,, Recent Advances on Elliptic and Parabolic Issues: Proceedings of the 2004 Swiss-Japanese Seminar, (2006), 6.

[14]

C. Hirota, T. Ishiwata and S. Yazaki, Numerical study and examples on singularities of solutions to anisotropic crystalline curvature flows of nonconvex polygonal curves,, Asymptotic analysis and singularitieselliptic and parabolic PDEs and related problems, 47 (2007), 543.

[15]

K. Ishii and H. M. Soner, Regularity and convergence of crystalline motion,, SIAM J. Math. Anal., 30 (1999), 19. doi: 10.1137/S0036141097317347.

[16]

T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena,, Japan Journal of Industrial and Applied Mathematics, 25 (2008), 233.

[17]

T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect,, Discrete Contin. Dyn. Syst., 4 (2011), 865. doi: 10.3934/dcdss.2011.4.865.

[18]

T. Ishiwata, Motion of Polygonal Curved Fronts by Crystalline Motion: V-Shaped Solutions and Eventual Monotonicity,, Discrete Contin. Dyn. Syst. 2011, (2011), 717.

[19]

T. Ishiwata and M. Tsutsumi, Semidiscretization in space of nonlinear degenerate parabolic equations with blow-up of the solution,, J. Comput. Math., 18 (2000), 571.

[20]

T. Ishiwata, T. K. Ushijima, H. Yagisita and S. Yazaki, Two examples of nonconvex self-similar solution curves for a crystalline curvature flow,, Proc. Japan Acad. Ser. A Math. Sci., 80 (2004), 151.

[21]

T. Ishiwata and S. Yazaki, On the blow-up rate for fast blow-up solutions arising in an anisotropic crystalline motion,, The Proceedings of the Sixth Japan-China Joint Seminar; a special issue of J. Comp. App. Math., 159 (2003), 55. doi: 10.1016/S0377-0427(03)00556-9.

[22]

M. Kimura, D. Tagami and S. Yazaki, Polygonal Hele-Shaw problem with surface tension,, Interfaces and Free Boundaries, 15 (2013), 77. doi: 10.4171/IFB/295.

[23]

R. Kobayashi and Y. Giga, On anisotropy and curvature effects for growing crystals,, Recent topics in mathematics moving toward science and engineering, 18 (2001), 207. doi: 10.1007/BF03168571.

[24]

N. Mizoguchi, Type-II blowup for a semilinear heat equation,, Adv. Differential Equations, 9 (2004), 1279.

[25]

N. Mizoguchi, Rate of type II blowup for a semilinear heat equation,, Math. Ann., 339 (2007), 839. doi: 10.1007/s00208-007-0133-z.

[26]

N. Mizoguchi, Blow-up rate of type II and the braid group theory,, Trans. Amer. Math. Soc., 363 (2011), 1419. doi: 10.1090/S0002-9947-2010-04784-1.

[27]

A. Stancu, Asymptotic behavior of solutions to a crystalline flow,, Hokkaido Math. J., 27 (1998), 303.

[28]

J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry,, Differential geometry, (1991), 321.

[29]

J. E. Taylor, J. W. Cahn and C. A. Handwerker, Geometric models of crystal growth,, Acta Metall. Mater., 40 (1992), 1443.

[30]

T. K. Ushijima and S. Yazaki, Convergence of a crystalline algorithm for the motion of a closed convex curve by a power of curvature $V=K^\alpha$,, SIAM J. Numer. Anal., 37 (2000), 500. doi: 10.1137/S0036142997330135.

[31]

T. K. Ushijima and S. Yazaki, Convergence of a crystalline approximation for an area-preserving motion,, J. Comp. App. Math., 166 (2004), 427. doi: 10.1016/j.cam.2003.08.041.

[32]

J. H. Wilkinson, The Algebraic Eigenvalue Problem,, Clarendon press, (1965).

[33]

T. Yamamoto, Sūchikaisekinyūmon (in Japanese),, Saiensu-sha (1976, (2003).

[34]

S. Yazaki, Asymptotic behavior of solutions to an expanding motion by a negative power of crystalline curvature,, Adv. Math. Sci. Appl., 12 (2002), 227.

[35]

S. Yazaki, Motion of nonadmissible convex polygons by crystalline curvature,, Publications of Research Institute for Mathematical Sciences, 43 (2007), 155.

[36]

S. Yazaki, An area-preserving crystalline curvature flow equation,, Topics in mathematical modeling, 4 (2008), 169.

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