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

doi:10.1007/s10473-024-0420-8

Abstract

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.

Key words: degenerate diffusion, $p$-Laplacian, traveling waves, stability

CLC Number:

35K57

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.

Trendmd

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