Journal of Statistical Physics

, Volume 172, Issue 6, pp 1663–1681 | Cite as

Front Propagation for Reaction–Diffusion Equations in Composite Structures

  • M. Freidlin
  • L. Koralov


We consider asymptotic problems concerning the motion of interface separating the regions of large and small values of the solution of a reaction–diffusion equation in the media consisting of domains with different characteristics (composites). Under certain conditions, the motion can be described by the Huygens principle in the appropriate Finsler (e.g., Riemannian) metric. In general, the motion of the interface has, in a sense, non-local nature. In particular, the interface may move by jumps. We are mostly concerned with the nonlinear term that is of KPP type. The results are based on limit theorems for large deviations.


Reaction–diffusion Large deviations Interface motion 

Mathematics Subject Classification

35K57 35A18 60F10 



While working on this article, M. Freidlin was supported by NSF Grant DMS-1411866 and L. Koralov was supported by ARO Grant W911NF1710419.


  1. 1.
    Azencott, R., Freidlin, M., Varadhan, S.R.S.: Large Deviations at Saint-Flour. Probability at Saint-Flour, p. viii+371. Springer, Heidelberg (2013)zbMATHGoogle Scholar
  2. 2.
    Barles, G., Evans, L.C., Souganidis, P.E.: Wavefronts propagation for reaction-diffusion systems. Duke Math. J. 68, 835–858 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Evans, L.C., Souganidis, P.E.: A PDE approach to geometric optics for certain semilinear parabolic equations. Indiana Univ. Math. J. 38, 141–172 (1989)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Freidlin, M.I.: Propagation of concentration waves due to a random motion connected with growth. Soviet Math. Dokl. 246, 544–548 (1979)Google Scholar
  5. 5.
    Freidlin, M.I.: Limit theorems for large deviations and reaction-diffusion equations. Ann. Probab. 13(3), 639–676 (1985)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Freidlin, M.I.: Functional Integration and Partial Differential Equations. Princeton University Press, Princeton (1985)zbMATHGoogle Scholar
  7. 7.
    Freidlin, M.I.: Coupled reaction-diffusion equations. Ann. Probab. 19(1), 29–57 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Freidlin, M.I., Hu, W.: Wave Front Propagation for reaction-diffusion equation in narrow random channels. Nonlinearity 26, 2333–2356 (2013)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Freidlin, M.I., Lee, T.Y.: Wave front propagation and large deviations for diffusion-transmutation processes. Probab. Theory Relat. Fields 106(1), 39–70 (1996)CrossRefGoogle Scholar
  10. 10.
    Freidlin, M.I., Spiliopolous, K.: Reation-diffusion equations with nonlinear boundary conditions in narrow domains. Asymptot. Anal. 59, 227–249 (2008)MathSciNetGoogle Scholar
  11. 11.
    Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, New York (2012)CrossRefGoogle Scholar
  12. 12.
    Freidlin, M.I., Gredeskul, S., Marchenko, A., Pastur, L., Hunter, J.: Wave Front Propagation for KPP-Type Equations. Surveys in Applied Mathematics, vol. 2, pp. 1–62. Plenum Press, Berlin (1995)Google Scholar
  13. 13.
    Gartner, J.: On large deviations from the invariant measure. Theory Probab. Appl. 22(1), 24–39 (1977)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Gartner, J.: Nonlinear diffusion equations and excitable media. Soviet Math. Dokl. 254(6), 1310–1314 (1980)MathSciNetGoogle Scholar
  15. 15.
    Liptser, R.: Large deviations for two scaled diffusions. Probab. Theory Relat. Fields 106, 71–104 (1996)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations