# American Institute of Mathematical Sciences

March  2013, 8(1): 379-395. doi: 10.3934/nhm.2013.8.379

## Traveling fronts of pyramidal shapes in competition-diffusion systems

 1 Center for Partial Differential Equations, East China Normal University, Minhang, Shanghai, 200241, China 2 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-38 Ookayama, Meguro-ku, Tokyo 152-8552

Received  January 2012 Revised  March 2013 Published  April 2013

It is well known that a competition-diffusion system has a one-dimensional traveling front. This paper studies traveling front solutions of pyramidal shapes in a competition-diffusion system in $\mathbb{R}^N$ with $N\geq 2$. By using a multi-scale method, we construct a suitable pair of a supersolution and a subsolution, and find a pyramidal traveling front solution between them.
Citation: Wei-Ming Ni, Masaharu Taniguchi. Traveling fronts of pyramidal shapes in competition-diffusion systems. Networks and Heterogeneous Media, 2013, 8 (1) : 379-395. doi: 10.3934/nhm.2013.8.379
##### References:
 [1] J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363. doi: 10.1007/s10884-011-9214-5. [2] J. K. Hale, "Ordinary Differential Equations," Wiley-Interscience, 1969. [3] F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069. [4] F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92. [5] M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. I. H. Poincaré, 23 (2006), 283-329. doi: 10.1016/j.anihpc.2005.03.003. [6] Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I. Singular perturbations, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 79-95. doi: 10.3934/dcdsb.2003.3.79. [7] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556. [8] Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343-349. doi: 10.1007/BF03167252. [9] Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253. [10] H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011. [11] H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819. [12] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, Berlin, 1984. doi: 10.1007/978-1-4612-5282-5. [13] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000. [14] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788. [15] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037. [16] M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011. [17] A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, RI}, 1994. [18] Z.-C. Wang, Traveling curved fronts in monotone bistable systems, Discrete Contin. Dyn. Syst., 32 (2012), 2339-2374. doi: 10.3934/dcds.2012.32.2339.

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##### References:
 [1] J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363. doi: 10.1007/s10884-011-9214-5. [2] J. K. Hale, "Ordinary Differential Equations," Wiley-Interscience, 1969. [3] F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069. [4] F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92. [5] M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. I. H. Poincaré, 23 (2006), 283-329. doi: 10.1016/j.anihpc.2005.03.003. [6] Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I. Singular perturbations, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 79-95. doi: 10.3934/dcdsb.2003.3.79. [7] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556. [8] Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343-349. doi: 10.1007/BF03167252. [9] Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253. [10] H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011. [11] H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819. [12] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, Berlin, 1984. doi: 10.1007/978-1-4612-5282-5. [13] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000. [14] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788. [15] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037. [16] M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011. [17] A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, RI}, 1994. [18] Z.-C. Wang, Traveling curved fronts in monotone bistable systems, Discrete Contin. Dyn. Syst., 32 (2012), 2339-2374. doi: 10.3934/dcds.2012.32.2339.
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