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2021, Volume 29, Issue 6: 3721-3740. doi: 10.3934/era.2021058
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Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations

  • College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China
  • 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(n≥4). 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

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  • 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(n≥4). 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.





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    [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.

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