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Traveling wave solutions to some reaction diffusion equations with fractional Laplacians

  • Changfeng GuiEmail author
  • Tingting Huan
Article

Abstract

We show the nonexistence of traveling wave solutions in the combustion model with fractional Laplacian \(\displaystyle (-\Delta )^s\) when \(\displaystyle s\in (0,1/2]\). Our method can be used to give a direct and simple proof of the nonexistence of traveling fronts for the usual Fisher-KPP nonlinearity. Also we prove the existence and nonexistence of traveling wave solutions for different ranges of the fractional power \(s\) for the generalized Fisher–KPP type model.

Mathematics Subject Classification

primary 35B32 35C07 35J20 35R09 35R11 45G05 47G10 

Notes

Acknowledgments

This work was partially supported by a grant from the Simons Foundation (Award # 199305) and a NSF IPA award. The authors would also like to thank the anonymous referee for helpful suggesions for the revision of the manuscript.

References

  1. 1.
    Aronson, D., Weinberger, H.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, partial differential equations and related topics, lecture notes in mathematics, pp. 4–49. Springer, Berlin (1975)zbMATHGoogle Scholar
  2. 2.
    Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and hamiltonian estimates. ArXiv (2010)Google Scholar
  3. 3.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cabré, X., Roquejoffre, J.: The influence of fractional diffusion in Fisher–KPP equations. Comm. Math. Phys. 320(3), 679–722 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 353–369 (1937)zbMATHGoogle Scholar
  6. 6.
    Gui, C., Zhao, M.: Traveling Wave Solutions of Allen–Cahn Equation with a Fractional Laplacian, Ann. I. H. Poincaré-AN (2014). doi: 10.1016/j.anihpc.2014.03.005
  7. 7.
    Kolmogorov, A., Petrovskii, I., Piskunov, N.: A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bull. Moscow Univ. Math. Ser. A 1, 1–25 (1937)Google Scholar
  8. 8.
    Landkof, N.S.: Foundations of Modern Potential Theory. In: Doohovskoy, A.P., (ed) Die Grundlehren der mathematischen Wissenschaften, Band 180. Translated from the Russian. Springer, New York, (1972)Google Scholar
  9. 9.
    Mellet, A., Roquejoffre, J., Sire, Y.: Existence and asymptotics of fronts in non local combustion models. Arxiv (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

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