Asymptotic behavior of spreading fronts in an anisotropic multi-stable equation on $ \mathit{\boldsymbol{\mathbb{R}^N}} $

Abstract

<p style='text-indent:20px;'>We consider the Cauchy problem for an anisotropic reaction diffusion equation with a multi-stable nonlinearity on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ N\ge 2 $\end{document}</tex-math></inline-formula> and investigate the large time behavior of the solution. This problem with a bistable nonlinearity has been investigated by Matano, Mori and Nara [<i>Ann. Inst. H. Poincaré Anal. Non Linéaire</i> 36 (2019), pp. 585-626]. They showed that under the suitable condition on the initial function the solution develops a spreading front whose position is closely approximated by the expanding Wulff shape for all large times. In this paper we will extend their results to the problem with a multi-stable type nonlinearity, that is, the case where the nonlinearity can be decomposed to <inline-formula><tex-math id="M4">\begin{document}$ K $\end{document}</tex-math></inline-formula> number of bistable nonlinearities and show that under certain conditions on the nonlinearity and the initial function the solution develops <inline-formula><tex-math id="M5">\begin{document}$ K $\end{document}</tex-math></inline-formula> number of spreading fronts whose positions are closely approximated by the expanding Wulff shapes with different expanding speeds. In other words, for any direction the solution on the ray of the direction looks like stacked traveling waves, that is, on the ray the solution approaches the so called propagating terrace. The key step for extension to multi-stable case is to construct <inline-formula><tex-math id="M6">\begin{document}$ K $\end{document}</tex-math></inline-formula> number of upper solutions and lower solutions all at once.</p>