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

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 \begin{document}$ L^{\infty} $\end{document} spaces on \begin{document}$ \mathbb{R}^{n}\; (n\geq4) $\end{document} . Secondly, we show that under general bounded perturbations (even very small), the pyramidal traveling fronts are not asymptotically stable by constructing a solution which oscillates permanently between two three-dimensional pyramidal traveling fronts on \begin{document}$ \mathbb{R}^{4} $\end{document} .