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

Denghui Wu; Zhen-Hui Bu; Denghui Wu; Zhen-Hui Bu

doi:10.3934/era.2021058

Electronic Research Archive

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

Denghui Wu ,
Zhen-Hui Bu ^,

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^{\infty} $ spaces on $ \mathbb{R}^{n}\; (n\geq4) $. 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 $ \mathbb{R}^{4} $.
- Reaction-diffusion equation,
- pyramidal traveling front,
- multidimensional stability,
- degenerate Fisher-KPP monostable nonlinearity,
- combustion nonlinearity
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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Abstract

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^{\infty} $ spaces on $ \mathbb{R}^{n}\; (n\geq4) $. 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 $ \mathbb{R}^{4} $.

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