Skip to main content
SpringerLink
Log in

Existence and stability of traveling waves for doubly degenerate diffusion equations

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper is concerned with the existence and stability of traveling waves for doubly degenerate diffusion equations, where the spatial diffusion operator is of the form \(\partial _x(|\partial _x u^m|^{p-2}\partial _x u^m)\) with \(m>0\) and \(p>1\). It is proved that, for the slow diffusion case \(m(p-1)>1\), there exists a minimum wave speed \(c^*\), such that the problem admits smooth traveling waves when wave speed \(c>c^*\) and semi-finite traveling waves with critical wave speed \(c=c^*\) while, for the fast diffusion case \(0<m(p-1)<1\), there is no nonnegative traveling wave solution. By the weighted energy method, we also show the \(L^1\)-stability of the traveling waves.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Aronson, D.G.: Density-dependent interaction-diffusion systems, In: Proceedings of the Advanced Seminar on Dynamics and Modeling of Reactive System. Academic Press, New York (1980)

  2. Aronson, D.G.: The porous medium equation. In: Fasano, A., Primicerio, M. (eds.) Some Problems in Nonlinear Diffusion, in Lecture Notes in Math., Springer-Verlag, New York/Berlin (1986)

  3. Atkinson, C., Reuter, G., Ridler-Rowe, C.: Traveling wave solution for some nonlinear diffusion equations. SIAM J. Math. Anal. 12, 880–892 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benguria, R.D., Depassier, M.C.: Variational characterization of the speed of reaction diffusion fronts for gradient dependent diffusion. Ann. Henri Poincaré 19, 2717–2726 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chern, I.-L., Mei, M., Yang, X., Zhang, Q.: Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay. J. Differ. Equ. 259, 1503–1541 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. DiBenedetto, E.: Degenerate Parabolic Equations, Universitext. Springer-Verlag, New York (1993)

    Book  MATH  Google Scholar 

  7. Enguica, R., Gavioli, A., Sanchez, L.: A class of singular first order differential equations with applications in reaction-diffusion. Discrete Contin. Dyn. Syst. 33, 173–191 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fisher, R.A.: The advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)

    Article  MATH  Google Scholar 

  9. Gurney, W.S.C., Blythe, S.P., Nisbet, R.M.: Nichlson’s blowflies revisited. Nature 287, 17–21 (1980)

    Article  Google Scholar 

  10. Gilding, B.H., Kersner, R.: A Fisher-KPP-type equation with density-dependent diffusion and convection: travelling wave solutions. J. Phys. A 38, 3367–3379 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang, R., Jin, C., Mei, M., Yin, J.: Existence and stability of traveling waves for degenerate reaction-diffusion equation with time delay. J. Nonlinear Sci. 28, 1011–1042 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang, R., Mei, M., Wang, Y.: Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete Contin. Dyn. Syst. 32(10), 3621–3649 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, R., Mei, M., Zhang, K., Zhang, Q.: Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete Contin. Dyn. Syst. 36(3), 1331–1353 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang, R., Tan, X., Yin, J.: The stability of curved fronts in a periodic shear flow. Appl. Math. Lett. 88(2), 33–40 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jin, C., Yin, J., Zheng, S.: Traveling waves for a time delayed newtonian filtration equation. J. Differ. Equ. 254, 1–29 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kolmogorov, A., Petrovskii, I., Piscounov, N.: Étude de l’équation de la diffusion avec croissance de la quantite de matière et son application à un problème biologique, Bull. Univ. Etat Moscou, Ser. Int. Sect. A Math. et Mecan., 1, 1–25 (1937)

  17. Li, W.T., Ruan, S.G., Wang, Z.C.: On the diffusive Nicholson’s blowflies equation with nonlocal delay. J. Nonlinear Sci. 17, 505–525 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liang, D., Wu, J.: travelling Waves and Numerical Approximations in a reaction advection diffusion equation with nonlocal delayed effects. J. Nonlinear Sci. 13, 289–310 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lin, C.-K., Lin, C.-T., Lin, Y., Mei, M.: Exponential stability of nonmonotone traveling waves for Nicholson’s blowflies equation. SIAM J. Math. Anal. 46, 1053–1084 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, G., Mei, M., Wong, Y.S.: Nonlinear stability of travelling wavefronts in an age-structured reaction-diffusion population model. Math. Biosci. Eng. 5(1), 85–100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lindqvist, P.: Notes on the p-Laplace Equation. University of Jyväskylä, Unversity Printing House (2006)

    MATH  Google Scholar 

  22. Liu, C.C., Mei, M., Yang, J.Q.: Global stability of traveling waves for nonlocal time-delayed degenerate diffusion equation. J. Differ. Equ. 306, 60–100 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ma, S.: Traveling waves for non-local delayed diffusion equation via auxiliary equation. J. Differ. Equ. 237, 259–277 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mei, M., Lin, C.K., Lin, C.T., So, J.W.-H.: Traveling wavefronts for time-delayed reaction-diffusion equation: (I) local nonlinearity. J. Differ. Equ. 247, 495–510 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mei, M., Ou, C., Zhao, X.-Q.: Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations. SIAM J. Math. Anal. 42, 2762–2790 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. De Pablo, A., Sánchez, A.: Travelling wave behaviour for a porous-Fisher equation. Eur. J. Pure Appl. Math. 9, 285–304 (1998)

    MathSciNet  MATH  Google Scholar 

  27. De Pablo, A., Vázquez, J.L.: Travelling waves and finite propagation in a reaction-diffusion equation. J. Differ. Equ. 93, 19–61 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. De Pablo, A., Vázquez, J.L.: The balance between strong reaction and slow diffusion. Comm. Part. Differ. Equ. 15, 159–183 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schaaf, K.W.: Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans. Am. Math. Soc. 302, 587–615 (1987)

    MathSciNet  MATH  Google Scholar 

  30. So, J.W.-H., Zou, X.: Traveling waves for the diffusive Nicholson’s blowflies equation. Appl. Math. Comp. 122, 385–392 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. So, J.W.-H., Wu, J., Zou, X.: A reaction-diffusion model for a single species with age structure: (I) traveling wavefronts on unbounded domains. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 457, 1841–1853 (2001)

    Article  MATH  Google Scholar 

  32. Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, vol. 33. OUP Oxford, Oxford (2006)

    Book  MATH  Google Scholar 

  33. Vázquez, J.L.: The Porous Medium Equation: Mathematical Theory. Oxford University Press, Oxford (2007)

    MATH  Google Scholar 

  34. Audrito, A., Vázquez, J.L.: The Fisher-KPP problem with doubly nonlinear diffusion. J. Differ. Equ. 263, 7647–7708 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wu, Z., Zhao, J., Yin, J., Li, H.: Nonlinear Diffusion Equations, World Scientific Publishing Co. Put. Ltd. (2001)

  36. Xu, T., Ji, S., Huang, R., Mei, M., Yin, J.: Theoretical and numerical studies on global stability of traveling waves with oscillations for time-delayed nonlocal dispersion equations. Int. J. Numer. Anal. Model. 17(1), 68–86 (2020)

    MathSciNet  MATH  Google Scholar 

  37. Xu, T., Ji, S., Mei, M., Yin, J.: Critical sharp front for doubly nonlinear degenerate diffusion equations with time delay. Nonlinearity 35(7), 3358–3384 (2022)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research of R. Huang was supported by NSFC Grant Nos. 11971179 and 12126204, GBABRF Grant Nos. 2020A1515010337, 2020B1515310005 and 2020B1515310013.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Rui Huang, Zhanghua Liang & Zhuangzhuang Wang

Authors
  1. Rui Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhanghua Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhuangzhuang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhuangzhuang Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, R., Liang, Z. & Wang, Z. Existence and stability of traveling waves for doubly degenerate diffusion equations. Z. Angew. Math. Phys. 74, 41 (2023). https://doi.org/10.1007/s00033-023-01938-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-023-01938-6

Keywords

Mathematics Subject Classification

Search

Navigation