Loading [MathJax]/jax/output/SVG/jax.js

Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations

  • Received: 01 April 2021 Revised: 01 July 2021 Published: 13 August 2021
  • Primary: 35K57, 35B35, 35C07; Secondary: 92D25

  • In this paper, multidimensional stability of pyramidal traveling fronts are studied to the reaction-diffusion equations with degenerate Fisher-KPP monostable and combustion nonlinearities. By constructing supersolutions and subsolutions coupled with the comparison principle, we firstly prove that under any initial perturbation (possibly large) decaying at space infinity, the three-dimensional pyramidal traveling fronts are asymptotically stable in weighted L spaces on Rn(n4). Secondly, we show that under general bounded perturbations (even very small), the pyramidal traveling fronts are not asymptotically stable by constructing a solution which oscillates permanently between two three-dimensional pyramidal traveling fronts on R4.

    Citation: Denghui Wu, Zhen-Hui Bu. Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations[J]. Electronic Research Archive, 2021, 29(6): 3721-3740. doi: 10.3934/era.2021058

    Related Papers:

    [1] Mengshi Shu, Rui Fu, Wendi Wang . A bacteriophage model based on CRISPR/Cas immune system in a chemostat. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1361-1377. doi: 10.3934/mbe.2017070
    [2] Miller Cerón Gómez, Eduardo Ibarguen Mondragon, Eddy Lopez Molano, Arsenio Hidalgo-Troya, Maria A. Mármol-Martínez, Deisy Lorena Guerrero-Ceballos, Mario A. Pantoja, Camilo Paz-García, Jenny Gómez-Arrieta, Mariela Burbano-Rosero . Mathematical model of interaction Escherichia coli and Coliphages. Mathematical Biosciences and Engineering, 2023, 20(6): 9712-9727. doi: 10.3934/mbe.2023426
    [3] Frédéric Mazenc, Gonzalo Robledo, Daniel Sepúlveda . A stability analysis of a time-varying chemostat with pointwise delay. Mathematical Biosciences and Engineering, 2024, 21(2): 2691-2728. doi: 10.3934/mbe.2024119
    [4] Gonzalo Robledo . Feedback stabilization for a chemostat with delayed output. Mathematical Biosciences and Engineering, 2009, 6(3): 629-647. doi: 10.3934/mbe.2009.6.629
    [5] Harry J. Dudley, Zhiyong Jason Ren, David M. Bortz . Competitive exclusion in a DAE model for microbial electrolysis cells. Mathematical Biosciences and Engineering, 2020, 17(5): 6217-6239. doi: 10.3934/mbe.2020329
    [6] Xiaomeng Ma, Zhanbing Bai, Sujing Sun . Stability and bifurcation control for a fractional-order chemostat model with time delays and incommensurate orders. Mathematical Biosciences and Engineering, 2023, 20(1): 437-455. doi: 10.3934/mbe.2023020
    [7] Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer . On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences and Engineering, 2007, 4(2): 319-338. doi: 10.3934/mbe.2007.4.319
    [8] Manel Dali Youcef, Alain Rapaport, Tewfik Sari . Study of performance criteria of serial configuration of two chemostats. Mathematical Biosciences and Engineering, 2020, 17(6): 6278-6309. doi: 10.3934/mbe.2020332
    [9] Alain Rapaport, Jérôme Harmand . Biological control of the chemostat with nonmonotonic response and different removal rates. Mathematical Biosciences and Engineering, 2008, 5(3): 539-547. doi: 10.3934/mbe.2008.5.539
    [10] Alexis Erich S. Almocera, Sze-Bi Hsu, Polly W. Sy . Extinction and uniform persistence in a microbial food web with mycoloop: limiting behavior of a population model with parasitic fungi. Mathematical Biosciences and Engineering, 2019, 16(1): 516-537. doi: 10.3934/mbe.2019024
  • In this paper, multidimensional stability of pyramidal traveling fronts are studied to the reaction-diffusion equations with degenerate Fisher-KPP monostable and combustion nonlinearities. By constructing supersolutions and subsolutions coupled with the comparison principle, we firstly prove that under any initial perturbation (possibly large) decaying at space infinity, the three-dimensional pyramidal traveling fronts are asymptotically stable in weighted L spaces on Rn(n4). Secondly, we show that under general bounded perturbations (even very small), the pyramidal traveling fronts are not asymptotically stable by constructing a solution which oscillates permanently between two three-dimensional pyramidal traveling fronts on R4.





    [1] Multidimensional nonlinear diffusions arising in population genetics. Adv. Math. (1978) 30: 33-76.
    [2] Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations I. Discrete Contin. Dyn. Syst. (2017) 37: 2395-2430.
    [3] Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations. Discrete Contin. Dyn. Syst. (2018) 38: 2251-2286.
    [4] Z.-H. Bu and Z.-C. Wang, Multidimensional stability of traveling fronts in combustion and non-KPP monostable equations, Z. Angew. Math. Phys., 69 (2018), Paper No. 12, 27 pp. doi: 10.1007/s00033-017-0906-5
    [5] Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete Contin. Dyn. Syst. Ser. B (2015) 20: 1015-1029.
    [6] Solutions of semilinear elliptic equations in RN with conical-shaped level sets. Comm. Partial Differential Equations (2000) 25: 769-819.
    [7] Existence and qualitative properties of multidimensional conical bistable fronts. Discrete Contin. Dyn. Syst. (2005) 13: 1069-1096.
    [8] Spatial decay and stability of traveling fronts for degenerate Fisher type equations in cylinder. J. Differential Equations (2018) 265: 5066-5114.
    [9] Multidimensional stability of planar traveling waves. Trans. Amer. Math. Soc. (1997) 349: 257-269.
    [10] The evolution of reaction-diffusion waves in a class of scalar reaction-diffusion equations: algebraic decay rates. Phys. D (2002) 167: 153-182.
    [11] Multidimensional stability of traveling waves in a bistable reaction-diffusion equation II. Comm. Partial Differential Equations (1992) 17: 1901-1924.
    [12] Stability of planar waves in mono-stable reaction-diffusion equations. Proc. Amer. Math. Soc. (2011) 139: 3611-3621.
    [13] Large time behavior of disturbed planar fronts in the Allen-Cahn equation. J. Differential Equations (2011) 251: 3522-3557.
    [14] Stability of planar waves in the Allen-Cahn equation. Comm. Partial Differential Equations (2009) 34: 976-1002.
    [15] Existence and global stability of traveling curved fronts in the Allen-Cahn equations. J. Differential Equations (2005) 213: 204-233.
    [16] Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation. Sci. China Math. (2013) 56: 1969-1982.
    [17] Traveling fronts of pyramidal shapes in the Allen-Cahn equations. SIAM J. Math. Anal. (2007) 39: 319-344.
    [18] The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations. J. Differential Equations (2009) 246: 2103-2130.
    [19] Nonplanar traveling fronts in reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearities. J. Differential Equations (2016) 260: 6405-6450.
    [20] Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I. Comm. Partial Differential Equations (1992) 17: 1889-1899.
    [21] Multidimensional stability of traveling fronts in monostable reaction-diffusion equations with complex perturbations. Sci. China Math. (2014) 57: 353-366.
    [22] Stability of planar travelling waves for bistable reaction-diffusion equations in multiple dimensions. Appl. Anal. (2014) 93: 653-664.
  • This article has been cited by:

    1. Xinzhi Ren, Xianning Liu, A competition un-stirred chemostat model with virus in an aquatic system, 2019, 98, 0003-6811, 2329, 10.1080/00036811.2018.1460811
    2. Wendi Wang, Rui Fu, Mengshi Shu, A bacteriophage model based on CRISPR/Cas immune system in a chemostat, 2017, 14, 1551-0018, 1361, 10.3934/mbe.2017070
    3. Saptarshi Sinha, Rajdeep K. Grewal, Soumen Roy, 2018, 103, 9780128151839, 103, 10.1016/bs.aambs.2018.01.005
    4. Saptarshi Sinha, Rajdeep Kaur Grewal, Soumen Roy, 2020, Chapter 18, 978-1-0716-0388-8, 309, 10.1007/978-1-0716-0389-5_18
    5. Sukhitha W. Vidurupola, Analysis of deterministic and stochastic mathematical models with resistant bacteria and bacteria debris for bacteriophage dynamics, 2018, 316, 00963003, 215, 10.1016/j.amc.2017.08.022
    6. Daniel A. Korytowski, Hal L. Smith, How nested and monogamous infection networks in host-phage communities come to be, 2015, 8, 1874-1738, 111, 10.1007/s12080-014-0236-6
    7. Saroj Kumar Sahani, Sunita Gakkhar, A Mathematical Model for Phage Therapy with Impulsive Phage Dose, 2020, 28, 0971-3514, 75, 10.1007/s12591-016-0303-0
    8. Sukhitha W. Vidurupola, Linda J. S. Allen, Impact of Variability in Stochastic Models of Bacteria-Phage Dynamics Applicable to Phage Therapy, 2014, 32, 0736-2994, 427, 10.1080/07362994.2014.889922
    9. WENDI WANG, DYNAMICS OF BACTERIA-PHAGE INTERACTIONS WITH IMMUNE RESPONSE IN A CHEMOSTAT, 2017, 25, 0218-3390, 697, 10.1142/S0218339017400010
    10. Hayriye Gulbudak, Paul L. Salceanu, Gail S. K. Wolkowicz, A delay model for persistent viral infections in replicating cells, 2021, 82, 0303-6812, 10.1007/s00285-021-01612-3
    11. Ei Ei Kyaw, Hongchan Zheng, Jingjing Wang, Htoo Kyaw Hlaing, Stability analysis and persistence of a phage therapy model, 2021, 18, 1551-0018, 5552, 10.3934/mbe.2021280
    12. Ei Ei Kyaw, Hongchan Zheng, Jingjing Wang, Stability and Hopf Bifurcation Analysis for a Phage Therapy Model with and without Time Delay, 2023, 12, 2075-1680, 772, 10.3390/axioms12080772
    13. Ei Ei Kyaw, Hongchan Zheng, Jingjing Wang, Hopf bifurcation analysis of a phage therapy model, 2023, 18, 2157-5452, 87, 10.2140/camcos.2023.18.87
    14. Zainab Dere, N.G. Cogan, Bhargav R. Karamched, Optimal control strategies for mitigating antibiotic resistance: Integrating virus dynamics for enhanced intervention design, 2025, 00255564, 109464, 10.1016/j.mbs.2025.109464
    15. Carli Peterson, Darsh Gandhi, Austin Carlson, Aaron Lubkemann, Emma Richardson, John Serralta, Michael S. Allen, Souvik Roy, Christopher M. Kribs, Hristo V. Kojouharov, A SIMPL Model of Phage-Bacteria Interactions Accounting for Mutation and Competition, 2025, 87, 0092-8240, 10.1007/s11538-025-01478-2
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2258) PDF downloads(134) Cited by(1)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog