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

Hiroshi Matsuzawa; Mitsunori Nara

doi:10.3934/dcds.2022069

Article Contents

2022, Volume 42, Issue 10: 4707-4740. Doi: 10.3934/dcds.2022069

This issue Previous Article Positive solutions to a nonlinear fractional equation with an external source term Next Article Classifications of positive solutions to an integral system involving the multilinear fractional integral inequality

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

Hiroshi Matsuzawa^1, , and
Mitsunori Nara^2,

1.
Department of Mathematics and Physics, Faculty of Science, Kanagawa University, 2946 Tsuchiya, Hiratsuka City, Kanagawa, 259-1293, Japan
2.
Faculty of Science and Engineering, Iwate University, Ueda 3-18-34, Morioka, Iwate, 020-8550, Japan

^*Corresponding author: Hiroshi Matsuzawa
^*Corresponding author: Hiroshi Matsuzawa

Received: September 2021

Revised: April 2022

Early access: May 2022

Published: October 2022

H. Matsuzawa was partially supported by Grant-in-Aid for Scientific Research (C) (20K03709), and M. Nara was partially supported by Grant-in-Aid for Scientific Research (C) (19K03556)

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Abstract

We consider the Cauchy problem for an anisotropic reaction diffusion equation with a multi-stable nonlinearity on $ \mathbb{R}^N $, $ N\ge 2 $ and investigate the large time behavior of the solution. This problem with a bistable nonlinearity has been investigated by Matano, Mori and Nara [Ann. Inst. H. Poincaré Anal. Non Linéaire 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 $ K $ number of bistable nonlinearities and show that under certain conditions on the nonlinearity and the initial function the solution develops $ K $ 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 $ K $ number of upper solutions and lower solutions all at once.

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Mathematics Subject Classification: Primary: 35C07, 35K57, 35B40; Secondary: 53E10.

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Figure 1. (Left) top view of a spreading front with $ K = 2 $, and the cross section of the solution that approaches at large time to the profile of a propagating terrace (stacked traveling waves) along the direction ${\boldsymbol{\nu}}$. (Right) A planar propagating terrace whose level set is tangential to the spreading Wulff shapes, and its cross section

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