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 & 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.  doi: 10.1007/s10884-011-9214-5.  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.  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.   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.  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.  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.  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.  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.  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.  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.  doi: 10.3934/dcds.2006.15.819.  Google Scholar

[12]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (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.   Google Scholar

[14]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations,, SIAM J. Math. Anal., 39 (2007), 319.  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.  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.  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 (1994).   Google Scholar

[18]

Z.-C. Wang, Traveling curved fronts in monotone bistable systems,, Discrete Contin. Dyn. Syst., 32 (2012), 2339.  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.  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.  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.   Google Scholar

[5]

M. Haragus and A. Scheel, Corner defects in almost planar interface propagation,, Ann. I. H. Poincaré, 23 (2006), 283.  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.  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.  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.  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.  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.  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.  doi: 10.3934/dcds.2006.15.819.  Google Scholar

[12]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (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.   Google Scholar

[14]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations,, SIAM J. Math. Anal., 39 (2007), 319.  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.  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.  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 (1994).   Google Scholar

[18]

Z.-C. Wang, Traveling curved fronts in monotone bistable systems,, Discrete Contin. Dyn. Syst., 32 (2012), 2339.  doi: 10.3934/dcds.2012.32.2339.  Google Scholar

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