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

Matthieu  Alfaro; Thomas  Giletti; Matthieu  Alfaro; Thomas  Giletti

doi:10.3934/nhm.2016001

Networks and Heterogeneous Media

2016, Volume 11, Issue 3: 369-393. doi: 10.3934/nhm.2016001

Previous Article Next Article

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

Matthieu Alfaro ^{1
,},
Thomas Giletti ^{2
,}

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

Received: 01 February 2015 Revised: 01 February 2016
Primary: 35K57; Secondary: 35B10.

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.
- Periodic media,
- monostable nonlinearity,
- ignition nonlinearity,
- pulsating traveling front,
- spreading properties
Citation: Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media[J]. Networks and Heterogeneous Media, 2016, 11(3): 369-393. doi: 10.3934/nhm.2016001

Related Papers:

Abstract

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.

References

[1]	M. Alfaro and T. Giletti, Asymptotic analysis of a monostable equation in periodic media, Tamkang J. Math., 47 (2016), 1-26.
[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, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975.
[3]	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
[4]	H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022
[5]	H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648. doi: 10.1002/cpa.21389
[6]	H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189. doi: 10.1016/j.jfa.2008.06.030
[7]	H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3
[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), 1101-1146. doi: 10.1016/j.matpur.2004.10.006
[9]	H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal., 16 (1985), 1207-1242. doi: 10.1137/0516088
[10]	H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497-572.
[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), 335-361.
[12]	R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x
[13]	F. Hamel, Qualitative properties of monostable pulsating fronts: Exponential decay and monotonicity, J. Math. Pures Appl., 89 (2008), 355-399. doi: 10.1016/j.matpur.2007.12.005
[14]	F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc., 13 (2011), 345-390. doi: 10.4171/JEMS/256
[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, 187-199, World Sci. Publ., River Edge, NJ, 1995.
[16]	J. I. Kanel, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, (Russian) Mat. Sb., 59 (1962), 245-288.
[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, Bjul. Moskowskogo Gos. Univ., 1937, 1-26.
[18]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. doi: 10.1142/3302
[19]	G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., (9) 92 (2009), 232-262. doi: 10.1016/j.matpur.2009.04.002
[20]	J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1021-1047. doi: 10.1016/j.anihpc.2009.02.003
[21]	P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Verlag, Basel, 2007.
[22]	arXiv:1503.09010.
[23]	N. Shigesada and K. Kawasaki, Biological Invasion: Theory and Practise, Oxford University Press, 1997.
[24]	A. Volpert, V. Volpert and V. Volpert, Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140, AMS Providence, RI, 1994.
[25]	H. Weinberger, On spreading speed and travelling waves for growth and migration, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3
[26]	J. Xin, Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity, J. Dynam. Differential Equations, 3 (1991), 541-573. doi: 10.1007/BF01049099
[27]	J. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Ration. Mech. Anal., 121 (1992), 205-233. doi: 10.1007/BF00410613
[28]	J. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, J. Statist. Phys., 73 (1993), 893-926. doi: 10.1007/BF01052815
[29]	J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296
[30]	A. Zlatoš, Generalized traveling waves in disordered media: Existence, uniqueness, and stability, Arch. Ration. Mech. Anal., 208 (2013), 447-480. doi: 10.1007/s00205-012-0600-x

Reader Comments

Your name:*

Email:*
© 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)