ON MONOTONE TRAVELING WAVES FOR NICHOLSON'S BLOWFLIES EQUATION WITH DEGENERATE $p$-LAPLACIAN DIFFUSION

  • Rui HUANG ,
  • Yong Wang ,
  • Zhuo YIN
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  • School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
E-mail: huangrui@m.scnu.edu.cn; YinZhuoMAX@163.com

Received date: 2023-06-15

  Online published: 2024-08-30

Supported by

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).

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.

Cite this article

Rui HUANG , Yong Wang , Zhuo YIN . ON MONOTONE TRAVELING WAVES FOR NICHOLSON'S BLOWFLIES EQUATION WITH DEGENERATE $p$-LAPLACIAN DIFFUSION[J]. Acta mathematica scientia, Series B, 2024 , 44(4) : 1550 -1571 . DOI: 10.1007/s10473-024-0420-8

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
[2] Atkinson C, Reuter G, Ridler-Rowe C. Traveling wave solution for some nonlinear diffusion equations. SIAM J Math Anal, 1981, 12: 880-892
[3] Audrito A. Bistable reaction equations with doubly nonlinear diffusion. Discrete Contin Dyn Syst, 2019, 39: 2977-3015
[4] Audrito A, Vázquez J L. The Fisher-KPP problem with doubly nonlinear "fast" diffusion. Nonlinear Anal, 2017, 157: 212-248
[5] Audrito A, Vázquez J L. The Fisher-KPP problem with doubly nonlinear diffusion. J Differential Equations, 2017, 263: 7647-7708
[6] Bramson M.Convergence of Solutions of the Kolmogorov Equation to Travelling Waves. Providence, RI: Mem Amer Math Soc, 1983
[7] Busenberg S, Iannelli M. A class of nonlinear diffusion problems in age-dependent population dynamics. Nonlinear Anal, 1983, 7: 501-529
[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
[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
[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
[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
[13] Fang J, Zhao X Q. Traveling waves for monotone semiflows with weak compactness. SIAM J Math Anal, 2014, 46: 3678-3704
[14] Fang J, Zhao X Q. Bistable traveling waves for monotone semiflows with applications. J Eur Math Soc, 2015, 17: 2243-2288
[15] Faria T, Trofimchuk S. Nonmonotone travelling waves in a single species reaction-diffusion equation with delay. J Differential Equations, 2006, 228: 357-376
[16] Fisher R A. The wave of advance of advantageous genes. Ann Eugen, 1937, 7: 335-369
[17] Gomez A, Trofimchuk S. Global continuation of monotone wavefronts. J London Math Soc, 2014, 89: 47-68
[18] Gurney W S C, Blythe S P, Nisbet R M. Nicholson's blowflies revisited. Nature, 1980, 287: 17-21
[19] Gurtin M E, MacCamy R C. On the diffusion of biological populations. Math Biosci, 1977, 33: 35-49
[20] Hamel F, Nadirashvili N. Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb R^N$. Arch Ration Mech Anal, 2001, 157: 91-163
[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
[22] Huang R, Mei M, Wang Y. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete Contin Dyn Syst, 2012, 32: 3621-3649
[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
[24] Huang R, Tan X, Yin J. The stability of curved fronts in a periodic shear flow. Appl Math Lett, 2019, 88: 33-40
[25] Huang R, Wang Z, Xu T.Smooth traveling waves for doubly nonlinear degenerate diffusion equations with time delay. Appl Anal, 2022. DOI: 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
[27] Jin C, Yin J, Zheng S. Traveling waves for a time delayed Newtonian filtration equation. J Differential Equations, 2013, 254: 1-29
[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 problème biologique. Bull Univ Etat Moscou, Ser Int, Sect A: Math et Mecan, 1937, 1: 1-25
[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
[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
[31] Ma S. Traveling waves for non-local delayed diffusion equation via auxiliary equation. J Differential Equations, 2007, 237: 259-277
[32] Mackey M C, Glass L. Oscillation and chaos in physiological control systems. Science, 1977, 197: 287-289
[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
[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
[35] Nicholson A J. An outline of the dynamics of animal population. Aust J Zool, 1954, 2: 9-65
[36] Schaaf K W. Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans Amer Math Soc, 1987, 302: 587-615
[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
[38] So J W H, Zou X. Traveling waves for the diffusive Nicholson's blowflies equation. Appl Math Comp, 2001, 122: 385-392
[39] Volpert A, Volpert Vi, Volpert Vl.Traveling Wave Solutions of Parabolic Systems. Transl Math Monogr. Providence RI: American Mathematical Society, 1994
[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
[41] Wu Z, Zhao J, Yin J, Li H. Nonlinear Diffusion Equations. Singapore: World Scientific, 2001
[42] Xin J. Front propagation in heterogeneous media. SIAM Rev, 2000, 42: 161-230
[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
[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
[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
[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
[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
[48] Yin J, Wang C. Evolutionary weighted $p$-Laplacian with boundary degeneracy. J Differential Equations, 2007, 237: 421-445
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