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May  2014, 34(5): 1775-1791. doi: 10.3934/dcds.2014.34.1775

## Bistable travelling waves for nonlocal reaction diffusion equations

 1 Univ. Montpellier 2, I3M, UMR CNRS 5149, CC051, Place Eugène Bataillon, 34095 Montpellier Cedex 5 2 INRA, Equipe BIOSP, Centre de Recherche d'Avignon, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9 3 CEFE, UMR 5175, CNRS, 1919 Route de Mende, 34293 Montpellier, France

Received  March 2013 Revised  August 2013 Published  October 2013

We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is "above and away", from the intermediate equilibrium. For focusing kernels we prove that, as expected, the wave connects 0 to 1. Our results also apply readily to the nonlocal ignition case.
Citation: Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775
##### References:
 [1] M. Alfaro and J. Coville, Rapid travelling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099. doi: 10.1016/j.aml.2012.05.006. [2] M. Alfaro, J. Coville and G. Raoul, Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait, to appear in Comm. Partial Differential Equations. doi: 10.1080/03605302.2013.828069. [3] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), Springer, Berlin,Lecture Notes in Math., 446, (1975), 5-49. [4] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. [5] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002. [6] H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal., 16 (1985), 1207-1242. doi: 10.1137/0516088. [7] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37. doi: 10.1007/BF01244896. [8] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497-572. [9] J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054. doi: 10.1088/0951-7715/24/11/002. [10] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361. [11] S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82. doi: 10.1051/mmnp:2006004. [12] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag: Berlin, 1977. [13] A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787. doi: 10.1016/j.jde.2010.11.011. [14] 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. [15] Ja. I. Kanel, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, (Russian) Mat. Sb., 59 (1962), 245-288. [16] O. A. Ladyženskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York, 1968. [17] G. Nadin, B. Perthame, L. Rossi and L. Ryzhik, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Mathematical Modelling of Natural Phenomena, 8 (2013), 33-41. doi: 10.1051/mmnp/20138304. [18] G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 553-557. doi: 10.1016/j.crma.2011.03.008. [19] A. Volpert, V. Volpert and V. Volpert, Travelling Wave Solutions of Parabolic Systems, Translated from the Russian manuscript by James F. Heyda. Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994. [20] Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025.

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##### References:
 [1] M. Alfaro and J. Coville, Rapid travelling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099. doi: 10.1016/j.aml.2012.05.006. [2] M. Alfaro, J. Coville and G. Raoul, Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait, to appear in Comm. Partial Differential Equations. doi: 10.1080/03605302.2013.828069. [3] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), Springer, Berlin,Lecture Notes in Math., 446, (1975), 5-49. [4] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. [5] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002. [6] H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal., 16 (1985), 1207-1242. doi: 10.1137/0516088. [7] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37. doi: 10.1007/BF01244896. [8] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497-572. [9] J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054. doi: 10.1088/0951-7715/24/11/002. [10] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361. [11] S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82. doi: 10.1051/mmnp:2006004. [12] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag: Berlin, 1977. [13] A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787. doi: 10.1016/j.jde.2010.11.011. [14] 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. [15] Ja. I. Kanel, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, (Russian) Mat. Sb., 59 (1962), 245-288. [16] O. A. Ladyženskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York, 1968. [17] G. Nadin, B. Perthame, L. Rossi and L. Ryzhik, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Mathematical Modelling of Natural Phenomena, 8 (2013), 33-41. doi: 10.1051/mmnp/20138304. [18] G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 553-557. doi: 10.1016/j.crma.2011.03.008. [19] A. Volpert, V. Volpert and V. Volpert, Travelling Wave Solutions of Parabolic Systems, Translated from the Russian manuscript by James F. Heyda. Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994. [20] Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025.
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