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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 & Continuous Dynamical Systems - A, 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. 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, 446 (1975), 5. [4] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. 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. 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. 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. doi: 10.1007/BF01244896. [8] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497. [9] J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation,, Nonlinearity, 24 (2011), 3043. 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. [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. doi: 10.1051/mmnp:2006004. [12] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Grundlehren der Mathematischen Wissenschaften, (1977). [13] A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation,, J. Differential Equations, 250 (2011), 1767. 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. 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. [16] O. A. Ladyženskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations,, translated from the Russian by Scripta Technica, (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. 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. 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, (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. doi: 10.1016/j.jde.2007.03.025.

show all references

##### 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. 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, 446 (1975), 5. [4] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. 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. 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. 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. doi: 10.1007/BF01244896. [8] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497. [9] J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation,, Nonlinearity, 24 (2011), 3043. 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. [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. doi: 10.1051/mmnp:2006004. [12] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Grundlehren der Mathematischen Wissenschaften, (1977). [13] A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation,, J. Differential Equations, 250 (2011), 1767. 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. 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. [16] O. A. Ladyženskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations,, translated from the Russian by Scripta Technica, (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. 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. 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, (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. doi: 10.1016/j.jde.2007.03.025.
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