New patterns of travelling waves in the generalized Fisher–Kolmogorov equation

Article
First Online:
Received:
Accepted:

Abstract

We prove the existence and uniqueness of a family of travelling waves in a degenerate (or singular) quasilinear parabolic problem that may be regarded as a generalization of the semilinear Fisher–Kolmogorov–Petrovski–Piscounov equation for the advance of advantageous genes in biology. Depending on the relation between the nonlinear diffusion and the nonsmooth reaction function, which we quantify precisely, we investigate the shape and asymptotic properties of travelling waves. Our method is based on comparison results for semilinear ODEs.

Keywords

Fisher–Kolmogorov equation Travelling waves Nonlinear diffusion Nonsmooth reaction function Comparison principle 

Mathematics Subject Classification

Primary 35Q92 35K92 Secondary 35K55 35K65 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Aronson D.G., Weinberger H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Drábek, P., Manásevich, R.F., Takáč, P.: Manifolds of critical points in a quasilinear model for phase transitions, In: Bonheure, D., Cuesta, M., Lami Dozo, E.J., Takáč, P., Van Schaftingen, J., Willem, M. (eds.) “Nonlinear elliptic partial differential equations”, Proceedings of the 2009 “International Workshop in Nonlinear Elliptic PDEs,” A celebration of Jean–Pierre Gossez’s 65-th birthday, September 2–4, 2009, Brussels. Contemporary Mathematics, vol. 540, pp. 95–134, American Mathematical Society, Providence (2011)Google Scholar
  3. 3.
    Enguiça R., Gavioli A., Sanchez L.: A class of singular first order differential equations with applications in reaction-diffusion. Discrete. Contin. Dyn. Syst. 33(1), 173–191 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Feireisl, E., Hilhorst, D., Petzeltová, H., Takáč, P.: Front propagation in nonlinear parabolic equations. J. London Math. Soc. 90(2), 551–572 (2014)Google Scholar
  5. 5.
    Fife P.C., McLeod J.B: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rational Mech. Anal. 65(4), 335–361 (1977)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fisher R.A.: The advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)CrossRefMATHGoogle Scholar
  7. 7.
    Hamel F., Nadirashvili N.: Travelling fronts and entire solutions of the Fisher–KPP equation in \({\mathbb{R}^{N}}\). Arch. Rational Mech. Anal. 157, 91–163 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hartman Ph.: Ordinary Differential Equations, 2nd edn. Birkhäuser, (1982)MATHGoogle Scholar
  9. 9.
    Hirsch, M.W.: Systems of differential equations that are competitive or cooperative II: convergence almost everywhere. SIAM J. Math. Anal. 16(3), 423–439 (1985, Online). doi: 10.1137/0516030
  10. 10.
    Il’yasov, Y. Sh., Takáč, P.: Optimal \({W^{2,2}_{\mathrm{loc}}}\)-regularity, Pohozhaev’s identity, and nonexistence of weak solutions to some quasilinear elliptic equations. J. Differ. Equ. 252, 2792–2822. (2012, Online). doi: 10.1016/j.jde.2011.10.020
  11. 11.
    Kolmogorov A., Petrovski I., Piscounov N.: Ètude de l’équation de la diffusion avec croissance de la quantité de la matière et son application à un problème biologique. Bull. Univ. Moskou Ser. Internat. Sec. A 1, 1–25 (1937)Google Scholar
  12. 12.
    Murray, J.D.: Mathematical Biology. In: Biomathematics Texts, vol. 19. Springer-Verlag, Berlin–Heidelberg-New York (1993)Google Scholar
  13. 13.
    Murray, J.D.: Mathematical biology I: an introduction, 3rd edn. In: Interdisciplinary Applied Mathematics, vol. 17. Springer-Verlag, Berlin–Heidelberg-New York (2002)Google Scholar
  14. 14.
    Turing A.M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952)CrossRefGoogle Scholar
  15. 15.
    Walter, W.: Ordinary differential equations. In: Grad. Texts in Math., vol. 182. Springer-Verlag, New York–Berlin–Heidelberg (1998)Google Scholar
  16. 16.
    Yi Q., Zhao J.-N.: Generation and propagation of interfaces for p-Laplacian equations. Acta Math. Sin. English Ser. 20(2), 319–332 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Mathematics, N.T.I.S. (Center of New Technologies for Information Society)University of West BohemiaPlzeňCzech Republic
  2. 2.Institut für MathematikUniversität RostockRostockGermany

Personalised recommendations