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May  2012, 17(3): 933-942. doi: 10.3934/dcdsb.2012.17.933

Spiral rotating waves of a geodesic curvature flow on the unit sphere

Bendong Lou 1,

1. 

Department of Mathematics, Tongji University, Shanghai 200092

Received  December 2010 Revised  July 2011 Published  January 2012

This paper is concerned with a geodesic curvature flow on the unit sphere. In each zone between the equator and the circle with latitude $\theta_0 \in (0, \frac{\pi}{2} ]$, we give the existence and uniqueness of a spiral rotating wave of the geodesic curvature flow.
Keywords: Geodesic curvature flow, Yin-Yang curve., spiral rotating wave.
Mathematics Subject Classification: Primary: 53D25, 58J35; Secondary: 34B05, 35K5.
Citation: Bendong Lou. Spiral rotating waves of a geodesic curvature flow on the unit sphere. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 933-942. doi: 10.3934/dcdsb.2012.17.933
References:
[1]

M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system, J. Differential Equations, 245 (2008), 505-565. doi: 10.1016/j.jde.2008.01.014.  Google Scholar

[2]

S. J. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom., 34 (1991), 491-514.  Google Scholar

[3]

F. Amdjadi and J. Gomatam, Spiral waves on static and moving spherical domains, J. Comput. Appl. Math., 182 (2005), 472-486. doi: 10.1016/j.cam.2004.12.027.  Google Scholar

[4]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[5]

K.-S. Chou and X.-P. Zhu, "The Curve Shorting Problem," Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.  Google Scholar

[6]

P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.  Google Scholar

[7]

J. Gomatam and F. Amdjadi, Reaction-diffusion equations on a sphere: Meandering of spiral waves, Physical Review E (3), 56 (1997), 3913-3919.  Google Scholar

[8]

R. Ikota, N. Ishimura and T. Yamaguchi, On the structure of steady solutions for the kinematic model of spiral waves in excitable media, Japan J. Indust. Appl. Math., 15 (1998), 317-330. doi: 10.1007/BF03167407.  Google Scholar

[9]

J. P. Keener, The core of the spiral, SIAM J. Appl. Math., 52 (1992), 1370-1390. doi: 10.1137/0152079.  Google Scholar

[10]

J. P. Keener and J. J. Tyson, Spiral waves in the Belousov-Zhabotinski reaction, Phys. D, 21 (1986), 307-324. doi: 10.1016/0167-2789(86)90007-2.  Google Scholar

[11]

B. Lou, Periodic rotating waves of a geodesic curvature flow on the sphere, Commun. Partial Differential Equations, 32 (2007), 525-541. doi: 10.1080/03605300701249663.  Google Scholar

[12]

B. D. Lou and L. Zhou, Singular limit of FitzHugh-Nagumo equations on a sphere, ZAMM Z. Angew. Math. Mech., 88 (2008), 644-649. doi: 10.1002/zamm.200700144.  Google Scholar

[13]

H. Matano, K.-I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568. doi: 10.3934/nhm.2006.1.537.  Google Scholar

[14]

K.-I. Nakamura, H. Matano, D. Hilhorst and R. Schätzle, Singular limit of a reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 95 (1999), 1165-1185. doi: 10.1023/A:1004518904533.  Google Scholar

[15]

T. Ogiwara and K.-I. Nakamura, Spiral traveling wave solutions of nonlinear diffusion equations related to a model of spiral crystal growth, Publ. Res. Inst. Math. Sci., 39 (2003), 767-783. doi: 10.2977/prims/1145476046.  Google Scholar

show all references

References:
[1]

M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system, J. Differential Equations, 245 (2008), 505-565. doi: 10.1016/j.jde.2008.01.014.  Google Scholar

[2]

S. J. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom., 34 (1991), 491-514.  Google Scholar

[3]

F. Amdjadi and J. Gomatam, Spiral waves on static and moving spherical domains, J. Comput. Appl. Math., 182 (2005), 472-486. doi: 10.1016/j.cam.2004.12.027.  Google Scholar

[4]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[5]

K.-S. Chou and X.-P. Zhu, "The Curve Shorting Problem," Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.  Google Scholar

[6]

P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.  Google Scholar

[7]

J. Gomatam and F. Amdjadi, Reaction-diffusion equations on a sphere: Meandering of spiral waves, Physical Review E (3), 56 (1997), 3913-3919.  Google Scholar

[8]

R. Ikota, N. Ishimura and T. Yamaguchi, On the structure of steady solutions for the kinematic model of spiral waves in excitable media, Japan J. Indust. Appl. Math., 15 (1998), 317-330. doi: 10.1007/BF03167407.  Google Scholar

[9]

J. P. Keener, The core of the spiral, SIAM J. Appl. Math., 52 (1992), 1370-1390. doi: 10.1137/0152079.  Google Scholar

[10]

J. P. Keener and J. J. Tyson, Spiral waves in the Belousov-Zhabotinski reaction, Phys. D, 21 (1986), 307-324. doi: 10.1016/0167-2789(86)90007-2.  Google Scholar

[11]

B. Lou, Periodic rotating waves of a geodesic curvature flow on the sphere, Commun. Partial Differential Equations, 32 (2007), 525-541. doi: 10.1080/03605300701249663.  Google Scholar

[12]

B. D. Lou and L. Zhou, Singular limit of FitzHugh-Nagumo equations on a sphere, ZAMM Z. Angew. Math. Mech., 88 (2008), 644-649. doi: 10.1002/zamm.200700144.  Google Scholar

[13]

H. Matano, K.-I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568. doi: 10.3934/nhm.2006.1.537.  Google Scholar

[14]

K.-I. Nakamura, H. Matano, D. Hilhorst and R. Schätzle, Singular limit of a reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 95 (1999), 1165-1185. doi: 10.1023/A:1004518904533.  Google Scholar

[15]

T. Ogiwara and K.-I. Nakamura, Spiral traveling wave solutions of nonlinear diffusion equations related to a model of spiral crystal growth, Publ. Res. Inst. Math. Sci., 39 (2003), 767-783. doi: 10.2977/prims/1145476046.  Google Scholar

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