May  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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[5]

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

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[29]

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

[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. Google Scholar

[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. Google Scholar

[32]

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

[33]

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

[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. Google Scholar

[35]

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

[36]

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

show all references

References:
[1]

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

[2]

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

[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. Google Scholar

[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. Google Scholar

[5]

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

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[29]

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

[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. Google Scholar

[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. Google Scholar

[32]

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

[33]

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

[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. Google Scholar

[35]

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

[36]

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

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