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On monotone traveling waves for Nicholson’s blowflies equation with degenerate P-Laplacian diffusion

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Abstract

We study the existence and stability of monotone traveling wave solutions of Nicholson’s blowflies equation with degenerate p-Laplacian diffusion. We prove the existence and nonexistence of non-decreasing smooth traveling wave solutions by phase plane analysis methods. Moreover, we show the existence and regularity of an original solution via a compactness analysis. Finally, we prove the stability and exponential convergence rate of traveling waves by an approximated weighted energy method.

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References

  1. Aronson D G. Density-dependent interaction-diffusion systems//Proc Adv Seminar on Dynamics and Modeling of Reactive System. New York: Academic Press, 1980

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Audrito A. Bistable reaction equations with doubly nonlinear diffusion. Discrete Contin Dyn Syst, 2019, 39: 2977–3015

    Article  MathSciNet  Google Scholar 

  4. Audrito A, Vázquez J L. The Fisher-KPP problem with doubly nonlinear “fast” diffusion. Nonlinear Anal, 2017, 157: 212–248

    Article  MathSciNet  Google Scholar 

  5. Audrito A, Vázquez J L. The Fisher-KPP problem with doubly nonlinear diffusion. J Differential Equations, 2017, 263: 7647–7708

    Article  MathSciNet  Google Scholar 

  6. Bramson M. Convergence of Solutions of the Kolmogorov Equation to Travelling Waves. Providence, RI: Mem Amer Math Soc, 1983

    Book  Google Scholar 

  7. Busenberg S, Iannelli M. A class of nonlinear diffusion problems in age-dependent population dynamics. Nonlinear Anal, 1983, 7: 501–529

    Article  MathSciNet  Google Scholar 

  8. Calvo J, Campos J, Caselles V, et al. Pattern formation in a flux limited reaction-diffusion equation of porous media type. Invent Math, 2016, 206: 57–108

    Article  MathSciNet  Google Scholar 

  9. Campos J, Guerrero P, Sánchez O, Soler J. On the analysis of traveling waves to a nonlinear flux limited reaction-diffusion equation. Ann Inst H Poincaré Anal Non Linéaire, 2013, 30: 141–155

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Du Y, Gárriz A, Quirós F. Travelling-wave behaviour in doubly nonlinear reaction-diffusion equations. arXiv: 2009.12959

  12. Fan X L, Zhang Q H. Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear Anal, 2003, 52: 1843–1852

    Article  MathSciNet  Google Scholar 

  13. Fang J, Zhao X Q. Traveling waves for monotone semiflows with weak compactness. SIAM J Math Anal, 2014, 46: 3678–3704

    Article  MathSciNet  Google Scholar 

  14. Fang J, Zhao X Q. Bistable traveling waves for monotone semiflows with applications. J Eur Math Soc, 2015, 17: 2243–2288

    Article  MathSciNet  Google Scholar 

  15. Faria T, Trofimchuk S. Nonmonotone travelling waves in a single species reaction-diffusion equation with delay. J Differential Equations, 2006, 228: 357–376

    Article  MathSciNet  Google Scholar 

  16. Fisher R A. The wave of advance of advantageous genes. Ann Eugen, 1937, 7: 335–369

    Article  Google Scholar 

  17. Gomez A, Trofimchuk S. Global continuation of monotone wavefronts. J London Math Soc, 2014, 89: 47–68

    Article  MathSciNet  Google Scholar 

  18. Gurney W S C, Blythe S P, Nisbet R M. Nicholson’s blowflies revisited. Nature, 1980, 287: 17–21

    Article  Google Scholar 

  19. Gurtin M E, MacCamy R C. On the diffusion of biological populations. Math Biosci, 1977, 33: 35–49

    Article  MathSciNet  Google Scholar 

  20. Hamel F, Nadirashvili N. Travelling fronts and entire solutions of the Fisher-KPP equation in ℝN. Arch Ration Mech Anal, 2001, 157: 91–163

    Article  MathSciNet  Google Scholar 

  21. 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, 2018, 28: 1011–1042

    Article  MathSciNet  Google Scholar 

  22. Huang R, Mei M, Wang Y. Planar traveling waves for nonlocal dispersion equation with monostable non-linearity. Discrete Contin Dyn Syst, 2012, 32: 3621–3649

    Article  MathSciNet  Google Scholar 

  23. 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, 2016, 36: 1331–1353

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Huang R, Wang Z, Xu T. Smooth traveling waves for doubly nonlinear degenerate diffusion equations with time delay. Appl Anal, 2022. DOI: https://doi.org/10.1080/00036811.2022.2136074

  26. Jin C, Yin J. Traveling wavefronts for a time delayed non-Newtonian filtration equation. Phys D, 2012, 241: 1789–1803

    Article  Google Scholar 

  27. Jin C, Yin J, Zheng S. Traveling waves for a time delayed Newtonian filtration equation. J Differential Equations, 2013, 254: 1–29

    Article  MathSciNet  Google Scholar 

  28. 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 probléme biologique. Bull Univ Etat Moscou, Ser Int, Sect A: Math et Mecan, 1937, 1: 1–25

    Google Scholar 

  29. Liang X, Zhao X Q. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm Pure Appl Math, 2007, 60: 1–40

    Article  MathSciNet  Google Scholar 

  30. 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, 2014, 46: 1053–1084

    Article  MathSciNet  Google Scholar 

  31. Ma S. Traveling waves for non-local delayed diffusion equation via auxiliary equation. J Differential Equations, 2007, 237: 259–277

    Article  MathSciNet  Google Scholar 

  32. Mackey M C, Glass L. Oscillation and chaos in physiological control systems. Science, 1977, 197: 287–289

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Nicholson A J. An outline of the dynamics of animal population. Aust J Zool, 1954, 2: 9–65

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  37. 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, 2001, 457: 1841–1853

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. Volpert A, Volpert Vi, Volpert Vl. Traveling Wave Solutions of Parabolic Systems. Transl Math Monogr. Rhode Island: American Mathematical Society, 1994

    Book  Google Scholar 

  40. Wang Y, Yin J, Wu Z. Periodic solutions of evolution p-laplacian equtions with nonlinear sources. J Math Anal Appl, 1998, 219: 76–96

    Article  MathSciNet  Google Scholar 

  41. Wu Z, Zhao J, Yin J, Li H. Nonlinear Diffusion Equations. Singapore: World Scientific, 2001

    Book  Google Scholar 

  42. Xin J. Front propagation in heterogeneous media. SIAM Rev, 2000, 42: 161–230

    Article  MathSciNet  Google Scholar 

  43. Xu T, Ji S, Huang R, et al. Theoretical and numerical studies on global stability of traveling waves with oscillations for time-delayed nonlocal dispersion equations. Int J Numer Anal Model, 2020, 17: 68–86

    MathSciNet  Google Scholar 

  44. Xu T, Ji S, Mei M, Yin J. Sharp oscillatory traveling waves of structured population dynamics model with degenerate diffusion. J Differential Equations, 2020, 269: 8882–8917

    Article  MathSciNet  Google Scholar 

  45. Xu T, Ji S, Mei M, Yin J. Traveling waves for time-delayed reaction diffusion equations with degenerate diffusion. J Differential Equations, 2018, 265: 4442–4485

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  47. Yin J, Jin C. Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form. Discrete Contin Dyn Syst Ser B, 2010, 13: 213–227

    MathSciNet  Google Scholar 

  48. Yin J, Wang C. Evolutionary weighted p-Laplacian with boundary degeneracy. J Differential Equations, 2007, 237: 421–445

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Rui Huang, Yong Wang & Zhuo Yin

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  1. Rui Huang
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  2. Yong Wang
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  3. Zhuo Yin
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Correspondence to Yong Wang.

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Conflict of Interest The authors declare no conflict of interest.

Additional information

Huang’s work was partially supported by the NSFC (11971179, 12371205); Wang’s work was partially supported by the National Key R&D Program of China (2021YFA1002900), the Guangdong Province Basic and Applied Basic Research Fund (2021A1515010235) and the Guangzhou City Basic and Applied Basic Research Fund (2024A04J6336).

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Huang, R., Wang, Y. & Yin, Z. On monotone traveling waves for Nicholson’s blowflies equation with degenerate P-Laplacian diffusion. Acta Math Sci 44, 1550–1571 (2024). https://doi.org/10.1007/s10473-024-0420-8

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  • DOI: https://doi.org/10.1007/s10473-024-0420-8

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