# American Institute of Mathematical Sciences

September  2016, 11(3): 369-393. doi: 10.3934/nhm.2016001

## Varying the direction of propagation in reaction-diffusion equations in periodic media

 1 IMAG, CC051, Université de Montpellier , 34095 Montpellier, France 2 IECL, Université de Lorraine, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France

Received  February 2015 Revised  February 2016 Published  August 2016

We consider a multidimensional reaction-diffusion equation of either ignition or monostable type, involving periodic heterogeneity, and analyze the dependence of the propagation phenomena on the direction. We prove that the (minimal) speed of the underlying pulsating fronts depends continuously on the direction of propagation, and so does its associated profile provided it is unique up to time shifts. We also prove that the spreading properties [25] are actually uniform with respect to the direction.
Citation: Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks & Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001
##### References:
 [1] M. Alfaro and T. Giletti, Asymptotic analysis of a monostable equation in periodic media,, Tamkang J. Math., 47 (2016), 1. Google Scholar [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, Partial differential equations and related topics (Program, (1974), 5. Google Scholar [3] 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. Google Scholar [4] H. Berestycki and F. Hamel, Front propagation in periodic excitable media,, Comm. Pure Appl. Math., 55 (2002), 949. doi: 10.1002/cpa.3022. Google Scholar [5] H. Berestycki and F. Hamel, Generalized transition waves and their properties,, Comm. Pure Appl. Math., 65 (2012), 592. doi: 10.1002/cpa.21389. Google Scholar [6] H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Funct. Anal., 255 (2008), 2146. doi: 10.1016/j.jfa.2008.06.030. Google Scholar [7] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence,, J. Math. Biol., 51 (2005), 75. doi: 10.1007/s00285-004-0313-3. Google Scholar [8] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating traveling fronts,, J. Math. Pures Appl., 84 (2005), 1101. doi: 10.1016/j.matpur.2004.10.006. Google Scholar [9] 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. Google Scholar [10] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497. Google Scholar [11] 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. Google Scholar [12] R. A. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar [13] F. Hamel, Qualitative properties of monostable pulsating fronts: Exponential decay and monotonicity,, J. Math. Pures Appl., 89 (2008), 355. doi: 10.1016/j.matpur.2007.12.005. Google Scholar [14] F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts,, J. Eur. Math. Soc., 13 (2011), 345. doi: 10.4171/JEMS/256. Google Scholar [15] W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media,, Boundary value problems for functional-differential equations, (1995), 187. Google Scholar [16] J. I. Kanel, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory,, (Russian) Mat. Sb., 59 (1962), 245. Google Scholar [17] A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Université d'Etat Moscou, (1937), 1. Google Scholar [18] G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co. Inc., (1996). doi: 10.1142/3302. Google Scholar [19] G. Nadin, Traveling fronts in space-time periodic media,, J. Math. Pures Appl., 92 (2009), 232. doi: 10.1016/j.matpur.2009.04.002. Google Scholar [20] J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1021. doi: 10.1016/j.anihpc.2009.02.003. Google Scholar [21] P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Verlag, (2007). Google Scholar [22] L. Rossi, The Freidlin-Gärtner formula for reaction term of any type,, , (). Google Scholar [23] N. Shigesada and K. Kawasaki, Biological Invasion: Theory and Practise,, Oxford University Press, (1997). Google Scholar [24] A. Volpert, V. Volpert and V. Volpert, Travelling Wave Solutions of Parabolic Systems,, Translations of Mathematical Monographs, (1994). Google Scholar [25] H. Weinberger, On spreading speed and travelling waves for growth and migration,, J. Math. Biol., 45 (2002), 511. doi: 10.1007/s00285-002-0169-3. Google Scholar [26] J. Xin, Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity,, J. Dynam. Differential Equations, 3 (1991), 541. doi: 10.1007/BF01049099. Google Scholar [27] J. Xin, Existence of planar flame fronts in convective-diffusive periodic media,, Arch. Ration. Mech. Anal., 121 (1992), 205. doi: 10.1007/BF00410613. Google Scholar [28] J. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media,, J. Statist. Phys., 73 (1993), 893. doi: 10.1007/BF01052815. Google Scholar [29] J. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161. doi: 10.1137/S0036144599364296. Google Scholar [30] A. Zlatoš, Generalized traveling waves in disordered media: Existence, uniqueness, and stability,, Arch. Ration. Mech. Anal., 208 (2013), 447. doi: 10.1007/s00205-012-0600-x. Google Scholar

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##### References:
 [1] M. Alfaro and T. Giletti, Asymptotic analysis of a monostable equation in periodic media,, Tamkang J. Math., 47 (2016), 1. Google Scholar [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, Partial differential equations and related topics (Program, (1974), 5. Google Scholar [3] 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. Google Scholar [4] H. Berestycki and F. Hamel, Front propagation in periodic excitable media,, Comm. Pure Appl. Math., 55 (2002), 949. doi: 10.1002/cpa.3022. Google Scholar [5] H. Berestycki and F. Hamel, Generalized transition waves and their properties,, Comm. Pure Appl. Math., 65 (2012), 592. doi: 10.1002/cpa.21389. Google Scholar [6] H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Funct. Anal., 255 (2008), 2146. doi: 10.1016/j.jfa.2008.06.030. Google Scholar [7] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence,, J. Math. Biol., 51 (2005), 75. doi: 10.1007/s00285-004-0313-3. Google Scholar [8] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating traveling fronts,, J. Math. Pures Appl., 84 (2005), 1101. doi: 10.1016/j.matpur.2004.10.006. Google Scholar [9] 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. Google Scholar [10] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497. Google Scholar [11] 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. Google Scholar [12] R. A. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar [13] F. Hamel, Qualitative properties of monostable pulsating fronts: Exponential decay and monotonicity,, J. Math. Pures Appl., 89 (2008), 355. doi: 10.1016/j.matpur.2007.12.005. Google Scholar [14] F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts,, J. Eur. Math. Soc., 13 (2011), 345. doi: 10.4171/JEMS/256. Google Scholar [15] W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media,, Boundary value problems for functional-differential equations, (1995), 187. Google Scholar [16] J. I. Kanel, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory,, (Russian) Mat. Sb., 59 (1962), 245. Google Scholar [17] A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Université d'Etat Moscou, (1937), 1. Google Scholar [18] G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co. Inc., (1996). doi: 10.1142/3302. Google Scholar [19] G. Nadin, Traveling fronts in space-time periodic media,, J. Math. Pures Appl., 92 (2009), 232. doi: 10.1016/j.matpur.2009.04.002. Google Scholar [20] J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1021. doi: 10.1016/j.anihpc.2009.02.003. Google Scholar [21] P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Verlag, (2007). Google Scholar [22] L. Rossi, The Freidlin-Gärtner formula for reaction term of any type,, , (). Google Scholar [23] N. Shigesada and K. Kawasaki, Biological Invasion: Theory and Practise,, Oxford University Press, (1997). Google Scholar [24] A. Volpert, V. Volpert and V. Volpert, Travelling Wave Solutions of Parabolic Systems,, Translations of Mathematical Monographs, (1994). Google Scholar [25] H. Weinberger, On spreading speed and travelling waves for growth and migration,, J. Math. Biol., 45 (2002), 511. doi: 10.1007/s00285-002-0169-3. Google Scholar [26] J. Xin, Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity,, J. Dynam. Differential Equations, 3 (1991), 541. doi: 10.1007/BF01049099. Google Scholar [27] J. Xin, Existence of planar flame fronts in convective-diffusive periodic media,, Arch. Ration. Mech. Anal., 121 (1992), 205. doi: 10.1007/BF00410613. Google Scholar [28] J. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media,, J. Statist. Phys., 73 (1993), 893. doi: 10.1007/BF01052815. Google Scholar [29] J. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161. doi: 10.1137/S0036144599364296. Google Scholar [30] A. Zlatoš, Generalized traveling waves in disordered media: Existence, uniqueness, and stability,, Arch. Ration. Mech. Anal., 208 (2013), 447. doi: 10.1007/s00205-012-0600-x. Google Scholar
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