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

  • * Corresponding author: Yoshihisa Morita

    * Corresponding author: Yoshihisa Morita

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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  • 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.

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

    Citation:

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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

    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

    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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