American Institute of Mathematical Sciences

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

Traveling fronts of pyramidal shapes in competition-diffusion systems

Wei-Ming Ni 1, and  Masaharu Taniguchi 2,

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.
Keywords: pyramidal shapes, Traveling front, competition-diffusion system..
Mathematics Subject Classification: Primary: 35C07; Secondary: 35K5.
Citation: Wei-Ming Ni, Masaharu Taniguchi. Traveling fronts of pyramidal shapes in competition-diffusion systems. Networks & 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.  Google Scholar

[2]

J. K. Hale, "Ordinary Differential Equations," Wiley-Interscience, 1969.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

J. K. Hale, "Ordinary Differential Equations," Wiley-Interscience, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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