Exponential Propagation for Fractional Reaction–Diffusion Cooperative Systems with Fast Decaying Initial Conditions

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Abstract

We study the time asymptotic propagation of solutions to the reaction–diffusion cooperative systems with fractional diffusion. We prove that the propagation speed is exponential in time, and we find the precise exponent of propagation. This exponent depends on the smallest index of the fractional laplacians and on the principal eigenvalue of the matrix DF(0) where F is the reaction term. We also note that this speed does not depend on the space direction.

Keywords

Fractional laplacian Nonlinear Fisher-KPP reaction–diffusion equation Cooperative systems Time asymptotic propagation 

Mathematics Subject Classification

Primary 35R11 Secondary 35B40 

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables. Dover Publications, New York (1972)MATHGoogle Scholar
  2. 2.
    Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusions arising in population genetics. Adv. Math. 30, 33–76 (1978)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Barles, G., Evans, L.C., Souganidis, P.E.: Wavefront propagation for reaction-diffusion systems of PDE. Duke Math. J. 61(3), 835–858 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bonforte, M., Vazquez, J.: Quantitative local and global a priori estimates for fractional nonlinear diffusion equations. Adv. Math. 250, 242–284 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Busca, J., Sirakov, B.: Harnack type estimates for nonlinear elliptic systems and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 543–590 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cabré, X., Roquejoffre, J.: The influence of fractional diffusion in Fisher-KPP equation. Commun. Math. Phys. 320, 679–722 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cabré, X., Coulon, A.C., Roquejoffre, J.M.: Propagation in Fisher-KPP type equations with fractional diffusion in periodic media. C. R. Math. Acad. Sci. Paris 350(19–20), 885–890 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Erdélyi, A.: Higher Transcendental Functions, vol. I. McGraw-Hill Book Company, Inc., New York (1953)MATHGoogle Scholar
  9. 9.
    Evans, L.C., Souganidis, P.E.: A PDE approach to geometric optics for certain semilinear parabolic equations. Indiana Univ. Math. J. 45(2), 141–172 (1989)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Felmer, P., Yangari, M.: Fast propagation for fractional KPP equations with slowly decaying initial conditions. SIAM J. Math. Anal. 45(2), 662–678 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hamel, F., Roques, L.: Fast propagation for KPP equations with slowly decaying initial conditions. J. Differ. Equ. 249, 1726–1745 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer-Verlag, New York (1981)Google Scholar
  13. 13.
    Kolmogorov, A.N., Petrovsky, I.G., Piskunov, N.S.: Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. État Moscou Sér. Inter. A 1, 1–26 (1937)Google Scholar
  14. 14.
    Lewis, M., Li, B., Weinberger, H.: Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 45, 219–233 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lui, R.: Biological growth and spread modeled by systems of recursions. I. Mathematical theory. Math. Biosci. 93(2), 269–295 (1989)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Stan, D., Vázquez, L.J.: The Fisher-KPP equation with nonlinear fractional diffusion. SIAM J. Math. Anal. 46(5), 3241–3276 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Weinberger, H.F., Lewis, M., Li, B.: Anomalous spreading speeds of cooperative recursion systems. J. Math. Biol. 55, 207–222 (2007)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Weinberger, H.F., Lewis, M., Li, B.: Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 183–218 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité Paul SabatierToulouse Cedex 4France
  2. 2.Departamento de MatemáticaEscuela Politécnica NacionalQuitoEcuador

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