Nonstandard Quasi-monotonicity: An Application to the Wave Existence in a Neutral KPP–Fisher Equation

  • Eduardo Hernández
  • Sergei TrofimchukEmail author


We revisit Wu and Zou non-standard quasi-monotonicity approach for proving existence of monotone wavefronts in monostable reaction–diffusion equations with delays. This allows to solve the problem of existence of monotone wavefronts in a neutral KPP–Fisher equation. In addition, using some new ideas proposed recently by Solar et al., we establish the uniqueness (up to a translation) of these monotone wavefronts.


Monostable equation Quasi-monotonicity Non-standard order Uniqueness KPP–Fisher delayed equation Neutral differential equation 

Mathematics Subject Classification

34K12 35K57 92D25 



This work was initiated during a research stay of S.T. at the São Paulo University at Ribeirão Preto, Brasil. It was supported by FAPESP (Brasil) Project 18/06658-1 and partially by FONDECYT (Chile) Project 1190712. The first author was supported by Fapesp (Brasil) Project 2017/13145-8. S.T. acknowledges the very kind hospitality of the DCM-USP and expresses his sincere gratitude to the Professors M. Pierri and E. Hernández for their support and hospitality.


  1. 1.
    Benguria, R., Solar, A.: An iterative estimation for disturbances of semi-wavefronts to the delayed Fisher–KPP equation. Proc. Am. Math. Soc. (2019). Google Scholar
  2. 2.
    Berestycki, H., Nadin, G., Perthame, B., Ryzhik, L.: The non-local Fisher–KPP equation: travelling waves and steady states. Nonlinearity 22, 2813–2844 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ducrot, A., Nadin, G.: Asymptotic behaviour of traveling waves for the delayed Fisher–KPP equation. J. Differ. Equ. 256, 3115–3140 (2014)CrossRefzbMATHGoogle Scholar
  4. 4.
    Fang, J., Zhao, X.-Q.: Monotone wavefronts of the nonlocal Fisher–KPP equation. Nonlinearity 24, 3043–3054 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gomez, A., Trofimchuk, S.: Monotone traveling wavefronts of the KPP-Fisher delayed equation. J. Differ. Equ. 250, 1767–1787 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gomez, A., Trofimchuk, S.: Global continuation of monotone wavefronts. J. Lond. Math. Soc. 89, 47–68 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkc. Ekvacioj 21, 11–41 (1978)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hernández, E., Trofimchuk, S.: Traveling wave solutions for partial neutral differential equations (2019) (submitted) Google Scholar
  9. 9.
    Hernández, E., Wu, J.: Traveling wave front for partial neutral differential equations. Proc. Am. Math. Soc. 146, 1603–1617 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huang, J., Zou, X.: Existence of traveling wavefronts of delayed reaction–diffusion systems without monotonicity. Discrete Continuous Dyn. Syst. 9, 925–936 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kwong, M.K., Ou, C.: Existence and nonexistence of monotone traveling waves for the delayed Fisher equation. J. Differ. Equ. 249, 728–745 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, W.-T., Wang, Z.-C.: Travelling fronts in diffusive and cooperative Lotka–Volterra system with nonlocal delays. ZAMP 58, 571–591 (2007)zbMATHGoogle Scholar
  13. 13.
    Liu, Y., Weng, P.: Asymptotic pattern for a partial neutral functional differential equation. J. Differ. Equ. 258, 3688–3741 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liu, Y.: Uniqueness of traveling wave solutions for a quasi-monotone reaction–diffusion equation with neutral type. Pure Math. 7(4), 310–321 (2017). CrossRefGoogle Scholar
  15. 15.
    Ma, S.: Traveling wavefronts for delayed reaction–diffusion systems via fixed point theorem. J. Differ. Equ. 171, 294–314 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Martin, R.H., Smith, H.L.: Abstract functional differential equations and reaction–diffusion systems. Trans. Am. Math. Soc. 321, 1–44 (1990)MathSciNetzbMATHGoogle Scholar
  17. 17.
    So, J.W.-H., Zou, X.: Traveling waves for the diffusive Nicholson’s blowfles equation. Appl. Math. Comput. 122, 385–392 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Solar, A., Trofimchuk, S.: A simple approach to the wave uniqueness problem. J. Differ. Equ. 266, 6647–6660 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Smith, H.L., Thieme, H.R.: Monotone semiflows in scalar non-quasimonotone functional differential equations. J. Math. Anal. Appl. 150, 289–306 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Smith, H.L., Thieme, H.R.: Strongly order preserving semiflows generated by functional differential equations. J. Differ. Equ. 93, 322–363 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Trofimchuk, E., Pinto, M., Trofimchuk, S.: Monotone waves for non-monotone and non-local monostable reaction–diffusion equations. J. Differ. Equ. 261, 1203–1236 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Trofimchuk, E., Alvarado, P., Trofimchuk, S.: On the geometry of wave solutions of a delayed reaction–diffusion equation. J. Differ. Equ. 246, 1422–1444 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Volpert, V., Trofimchuk, S.: Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities. to appear (2019)Google Scholar
  24. 24.
    Wang, Z.-C., Li, W.-T., Ruan, S.: Travelling wave fronts in reaction–diffusion systems with spatio-temporal delays. J. Differ. Equ. 222, 185–232 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, Z.-C., Li, W.-T.: Monotone travelling fronts of a food-limited population model with nonlocal delay. Nonlinear Anal. Real World Appl. 8, 699–712 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wu, J., Zou, X.: Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations. J. Differ. Equ. 135, 315–357 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wu, J., Zou, X.: Traveling wave fronts of reaction–diffusion systems with delay. J. Dyn. Differ. Equ. 13, 651–687 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wu, J., Zou, X.: Erratum to “Traveling wave fronts of reaction–diffusion systems with delays” [J . Dyn. Differ. Equ. 13, 651, 687 (2001)]. J. Dyn. Differ. Equ. 20, 531–533 (2008)Google Scholar

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Authors and Affiliations

  1. 1.Departamento de Computação e Matemática, Faculdade de Filosofia, Ciências e Letras de Ribeirão PretoUniversidade de São PauloRibeirão PretoBrazil
  2. 2.Instituto de Matemática y FísicaUniversidad de TalcaTalcaChile

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