# American Institute of Mathematical Sciences

December  2021, 29(6): 3721-3740. doi: 10.3934/era.2021058

## 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

* Corresponding author: Zhen-Hui Bu

Received  April 2021 Revised  July 2021 Published  December 2021 Early access  August 2021

Fund Project: This paper was supported by Natural Science Basic Research Program in Shaanxi Province of China (2020JQ-236, 2019JQ-246)

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 $\mathbb{R}^{4}$.

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