Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph

Shuichi Jimbo; Yoshihisa Morita

doi:10.3934/dcds.2021026

Article Contents

2021, Volume 41, Issue 9: 4013-4039. Doi: 10.3934/dcds.2021026 open access

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Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph

Shuichi Jimbo^1, and
Yoshihisa Morita^2, ,

1.
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
2.
Department of Applied Mathematics and Informatics, Ryukoku University, Seta Otsu 520-2194, Japan

^* Corresponding author: Yoshihisa Morita
^* Corresponding author: Yoshihisa Morita

Received: May 31, 2020

Revised: November 30, 2020

Early access: January 2021

Published: September 2021

The authors were partially supported by JSPS KAKENHI Grant Number JP18H01139 and the first author was supported in part by JSPS KAKENHI Grant Number JP16K05218

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Abstract

We deal with the bistable reaction-diffusion equation in an infinite star graph, which consists of several half-lines with a common end point. The aim of our study is to show the existence of front-like entire solutions together with the asymptotic behaviors as $ t\to\pm\infty $ and classify the entire solutions according to their behaviors, where an entire solution is meant by a classical solution defined for all $ t\in(-\infty, \infty) $. To this end, we give a condition under that the front propagation is blocked by the emergence of standing stationary solutions. The existence of an entire solution which propagates beyond the blocking is also shown.

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Mathematics Subject Classification: Primary: 35K57, 35B08; Secondary: 35B35, 35B40.

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Figure 1. An example of the domain $ \Omega $ consisting of $ \Omega_k\; (1\leq k\leq5) $. $ \Omega_i\; (i = 1,2) $ are wave incoming branches, where the arrows in the figure indicate the directions of the front propagation

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Figure 2. Simplified dynamics around the stationary solutions for the cases $ (a)\; F(1)+(\kappa^2-1)F(a)<0,\; (b)\; F(1)+(\kappa^2-1)F(a) = 0, $ and $ (c)\; F(1)+(\kappa^2-1)F(a)>0 $. Although the phase space is infinite-dimensional, the figures indicate one-dimensional dynamics on the dominant subspace. $ \hat{u}, \hat{u}^{(1)}, \hat{u}^{(2)} $ and $ \hat{u}^{(12)} $ stand for the entire solutions stated in Theorem 1.1

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Figure 3. Case: $ m = 4,\; d_1 = d_2 = d_3 = d_4. $ (a) $ \kappa = 9 $, and $ F(1)+(\kappa^2-1)F(a)<0 $. An incoming front in $ \Omega_1 $ is blocked. (b) $ \kappa^+ = 0 $, so $ F(1)+((\kappa^+)^2-1)F(a)>0 $. Two incoming fronts from $ \Omega_1 $ and $ \Omega_2 $ go through the junction and propagate to the branches $ \Omega_3 $ and $ \Omega_4 $

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