Traveling waves in a one-dimensional heterogeneous medium

James Nolen; Lenya Ryzhik

Annales de l'I.H.P. Analyse non linéaire (2009)

  • Volume: 26, Issue: 3, page 1021-1047
  • ISSN: 0294-1449

How to cite


Nolen, James, and Ryzhik, Lenya. "Traveling waves in a one-dimensional heterogeneous medium." Annales de l'I.H.P. Analyse non linéaire 26.3 (2009): 1021-1047. <>.

author = {Nolen, James, Ryzhik, Lenya},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {equation of ignition type; ergodic reaction rate; long time limit; random traveling waves},
language = {eng},
number = {3},
pages = {1021-1047},
publisher = {Elsevier},
title = {Traveling waves in a one-dimensional heterogeneous medium},
url = {},
volume = {26},
year = {2009},

AU - Nolen, James
AU - Ryzhik, Lenya
TI - Traveling waves in a one-dimensional heterogeneous medium
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2009
PB - Elsevier
VL - 26
IS - 3
SP - 1021
EP - 1047
LA - eng
KW - equation of ignition type; ergodic reaction rate; long time limit; random traveling waves
UR -
ER -


  1. [1] Angenent S.B., The zero set of a solution of a parabolic equation, J. Reine Angew. Math.390 (1988) 79-96. Zbl0644.35050MR953678
  2. [2] Berestycki H., Hamel F., Fronts in periodic excitable media, Comm. Pure Appl. Math.60 (2002) 949-1032. Zbl1024.37054MR1900178
  3. [3] Berestycki H., Hamel F., Fronts and invasions in general domains, C. R. Acad. Sci. Paris Ser. I343 (2006) 711-716. Zbl1103.35044MR2284698
  4. [4] Berestycki H., Hamel F., Generalized travelling waves for reaction–diffusion equations, in: Perspectives in Nonlinear Partial Differential Equations. In Honor of H. Brezis, Contemp. Math., vol. 46, Amer. Math. Soc., 2007, pp. 101-123. Zbl1200.35169MR2373726
  5. [5] Berestycki H., Nirenberg L., On the method of moving planes and the sliding method, Bull. Braz. Math. Soc. (N.S.)22 (1991) 1-37. Zbl0784.35025MR1159383
  6. [6] Fife P.C., McLeod J.B., The approach of solutions of nonlinear diffusion equations by traveling front solutions, Arch. Ration. Mech. Anal.65 (1977) 335-361. Zbl0361.35035MR442480
  7. [7] Freidlin M., Gärtner J., On the propagation of concentration waves in periodic and random media, Soviet Math. Dokl.20 (1979) 1282-1286. Zbl0447.60060MR553200
  8. [8] Freidlin M., Functional Integration and Partial Differential Equations, Ann. of Math. Stud., vol. 109, Princeton Univ. Press, Princeton, NJ, 1985. Zbl0568.60057MR833742
  9. [9] Freidlin M., Limit theorems for large deviations and reaction–diffusion equations, Ann. Probab.13 (1985) 639-675. Zbl0576.60070MR799415
  10. [10] Freidlin M., Geometric optics approach to reaction–diffusion equations, SIAM J. Appl. Math.46 (1986) 222-232. Zbl0626.35047MR833475
  11. [11] Freidlin M., Reaction–diffusion in incompressible fluid: Asymptotic problems, J. Differential Equations179 (2002) 44-96. Zbl1043.35079MR1883738
  12. [12] Hamel F., Nadirashvili N., Entire solutions of the KPP equation, Comm. Pure Appl. Math.52 (1999) 1255-1276. Zbl0932.35113MR1699968
  13. [13] Hamel F., Nadirashvili N., Travelling fronts and entire solutions of the Fisher–KPP equation in R N , Arch. Ration. Mech. Anal.157 (2001) 91-163. Zbl0987.35072MR1830037
  14. [14] Kanel Ya., Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sb.59 (1962) 245-288. MR157130
  15. [15] Kosygina E., Rezakhanlou F., Varadhan S.R.S., Stochastic homogenization of Hamilton–Jacobi–Bellman equations, Comm. Pure Appl. Math.59 (2006) 1489-1521. Zbl1111.60055MR2248897
  16. [16] Krylov N.V., Safonov M.V., A property of the solutions of parabolic equations with measureable coefficients, (English translation), Izv. Akad. Nauk SSSR Ser. Mat.16 (1) (1981) 151-164. Zbl0464.35035MR563790
  17. [17] Lewis T.J., Keener J.P., Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math.61 (2000) 293-316. Zbl0967.35067MR1776397
  18. [18] Liggett T.M., An improved subadditive ergodic theorem, Ann. Probab.13 (1985) 1279-1285. Zbl0579.60023
  19. [19] Lions P.-L., Souganidis P., Homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media, Comm. Partial Differential Equations30 (2005) 335-375. Zbl1065.35047
  20. [20] Majda A., Souganidis P., Large scale front dynamics for turbulent reaction–diffusion equations with separated velocity scales, Nonlinearity7 (1994) 1-30. Zbl0839.76093
  21. [21] Mallordy J.-F., Roquejoffre J.-M., A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal.26 (1995) 1-20. Zbl0813.35041
  22. [22] H. Matano, talks presented at various conferences. 
  23. [23] A. Mellet, J.-M. Roquejoffre, Y. Sire, Generalized fronts for one-dimensional reaction–diffusion equations, preprint, 2008. Zbl1180.35294MR2552789
  24. [24] Nolen J., Xin J., Variational principle of KPP front speeds in temporally random shear flows with applications, Comm. Math. Phys.269 (2007) 493-532. Zbl1114.35095MR2274555
  25. [25] J. Nolen, J. Xin, Asymptotic spreading of KPP reactive fronts in incompressible space–time random flows, Ann. Inst. H. Poincaré Anal. Non Linéaire (2008), in press, 10.1016/j.anihpc.2008.02.005. Zbl1177.35172MR2526403
  26. [26] Roquejoffre J.-M., Eventual monotonicity and convergence to traveling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré14 (4) (1997) 499-552. Zbl0884.35013MR1464532
  27. [27] Shen W., Traveling waves in diffusive random media, J. Dynam. Differential Equations16 (4) (2004) 1011-1060. Zbl1082.35081MR2110054
  28. [28] Weinberger H., On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol.45 (2002) 511-548. Zbl1058.92036MR1943224
  29. [29] Xin J., Existence of planar flame fronts in convective–diffusive periodic media, Arch. Ration. Mech. Anal.121 (1992) 205-233. Zbl0764.76074MR1188981
  30. [30] Xin J., Existence and nonexistence of traveling waves and reaction–diffusion front propagation in periodic media, J. Stat. Phys.73 (1993) 893-926. Zbl1102.35340MR1251222

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