# American Institute of Mathematical Sciences

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

 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.
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. doi: 10.1016/j.jde.2008.01.014. [2] S. J. Altschuler, Singularities of the curve shrinking flow for space curves,, J. Differential Geom., 34 (1991), 491. [3] F. Amdjadi and J. Gomatam, Spiral waves on static and moving spherical domains, J., Comput. Appl. Math., 182 (2005), 472. doi: 10.1016/j.cam.2004.12.027. [4] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential Equations, 96 (1992), 116. doi: 10.1016/0022-0396(92)90146-E. [5] K.-S. Chou and X.-P. Zhu, "The Curve Shorting Problem,", Chapman & Hall/CRC, (2001). doi: 10.1201/9781420035704. [6] P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces,", CBMS-NSF Regional Conference Series in Applied Mathematics, 53 (1988). [7] J. Gomatam and F. Amdjadi, Reaction-diffusion equations on a sphere: Meandering of spiral waves,, Physical Review E (3), 56 (1997), 3913. [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. doi: 10.1007/BF03167407. [9] J. P. Keener, The core of the spiral,, SIAM J. Appl. Math., 52 (1992), 1370. doi: 10.1137/0152079. [10] J. P. Keener and J. J. Tyson, Spiral waves in the Belousov-Zhabotinski reaction,, Phys. D, 21 (1986), 307. doi: 10.1016/0167-2789(86)90007-2. [11] B. Lou, Periodic rotating waves of a geodesic curvature flow on the sphere,, Commun. Partial Differential Equations, 32 (2007), 525. doi: 10.1080/03605300701249663. [12] B. D. Lou and L. Zhou, Singular limit of FitzHugh-Nagumo equations on a sphere,, ZAMM Z. Angew. Math. Mech., 88 (2008), 644. doi: 10.1002/zamm.200700144. [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. doi: 10.3934/nhm.2006.1.537. [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. doi: 10.1023/A:1004518904533. [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. doi: 10.2977/prims/1145476046.

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##### 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. doi: 10.1016/j.jde.2008.01.014. [2] S. J. Altschuler, Singularities of the curve shrinking flow for space curves,, J. Differential Geom., 34 (1991), 491. [3] F. Amdjadi and J. Gomatam, Spiral waves on static and moving spherical domains, J., Comput. Appl. Math., 182 (2005), 472. doi: 10.1016/j.cam.2004.12.027. [4] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential Equations, 96 (1992), 116. doi: 10.1016/0022-0396(92)90146-E. [5] K.-S. Chou and X.-P. Zhu, "The Curve Shorting Problem,", Chapman & Hall/CRC, (2001). doi: 10.1201/9781420035704. [6] P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces,", CBMS-NSF Regional Conference Series in Applied Mathematics, 53 (1988). [7] J. Gomatam and F. Amdjadi, Reaction-diffusion equations on a sphere: Meandering of spiral waves,, Physical Review E (3), 56 (1997), 3913. [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. doi: 10.1007/BF03167407. [9] J. P. Keener, The core of the spiral,, SIAM J. Appl. Math., 52 (1992), 1370. doi: 10.1137/0152079. [10] J. P. Keener and J. J. Tyson, Spiral waves in the Belousov-Zhabotinski reaction,, Phys. D, 21 (1986), 307. doi: 10.1016/0167-2789(86)90007-2. [11] B. Lou, Periodic rotating waves of a geodesic curvature flow on the sphere,, Commun. Partial Differential Equations, 32 (2007), 525. doi: 10.1080/03605300701249663. [12] B. D. Lou and L. Zhou, Singular limit of FitzHugh-Nagumo equations on a sphere,, ZAMM Z. Angew. Math. Mech., 88 (2008), 644. doi: 10.1002/zamm.200700144. [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. doi: 10.3934/nhm.2006.1.537. [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. doi: 10.1023/A:1004518904533. [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. doi: 10.2977/prims/1145476046.
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