The dynamics of pulse solutions for reaction diffusion systems on a star shaped metric graph with the Kirchhoff's boundary condition

Shin-Ichiro Ei; Ken Mitsuzono; Haruki Shimatani

doi:10.3934/dcdsb.2022209

Article Contents

2023, Volume 28, Issue 12: 6064-6091. Doi: 10.3934/dcdsb.2022209 open access

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The dynamics of pulse solutions for reaction diffusion systems on a star shaped metric graph with the Kirchhoff's boundary condition

1.
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan

^*Corresponding author: Shin-Ichiro Ei
^*Corresponding author: Shin-Ichiro Ei

Dedicated to the memory of Professor Masayasu Mimura

Received: April 2022

Revised: September 2022

Early access: October 2022

Published: December 2023

The first author was supported in part by JST CREST (No. JPMJCR14D3) and JSPS KAKENHI Grant Numbers JP22H01129

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Abstract

In this paper, we consider motions of localized patterns for reaction-diffusion systems of general types on a metric star graph which consists of several half-lines with a common end point called "the junction point", where the Kirchhoff boundary condition is imposed. Assuming the existence and the stability of pulse and front like patterns for corresponding 1dimensional problems of reaction-diffusion systems, we rigorously derive ordinary differential equations describing the motions of them on a metric star graph. As the application, the attractive motion of a single pulse solution for the Gray-Scott model toward the junction point is shown. It is also shown that a single front solution of Allen-Cahn equation is repulsive against the junction point. The motion of multi pulse solutions and front solutions are also treated.

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Mathematics Subject Classification: Primary: 35B32, 35B35, 35K55, 35K57; Secondary: 35Q92.

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Figure 1. Schematic graph of a metric star graph in the case that $ R = 3 $, so called "Y-shaped metric graph"

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Figure 2. The flows for the ODE (19). (A): Flows in the case 1-a) are drawn for $ R = 5 $. Each flow corresponds to (a): $ l_1(0) = l_2(0) $, (b): $ 0 < l_1(0) - l_2(0) < |\frac{1}{\alpha}\log\frac{2}{R-2}| $, (c): $ 0 < l_2(0) - l_1(0) < |\frac{1}{\alpha}\log\frac{2}{R-2}| $. (B): Flows in the case 1-c) ($ R = 3 $) are drawn. Each flow corresponds to (a): $ l_1(0) = l_2(0) $, (b): $ 0 < l_1(0) - l_2(0) < \frac{\log 2}{\alpha} $, (c): $ 0 < l_2(0) - l_1(0) < \frac{\log 2}{\alpha} $

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Figure 3. Movement of a single front solution for (23) by a numerical simulation. (A) denotes the location $ l_1(t) $ of the zero level point of the front solution. (B) draws the time evolution of $ l_1(t) $. The movement apart from $ O $ is observed. In the numerical simulations, each length of $ \Omega_j $ is taken sufficiently large more than 10 with the Neumann boundary condition in the opposite side of $ O $. The same treatments are done for other figures of numerical simulations as in Figures 4, 5, and 6

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Figure 4. Movement of two front solutions for (23) in the case when $ R = 5 $ and $ r = 2 $ corresponding to 2-a) when $ l_2(0) - l_1(0) > \frac{1}{\alpha}\log\frac{2}{3} $ is satisfied. (A) denotes the locations $ l_1(t) $, $ l_2(t) $ of the zero level points of the front solutions on $ \Omega_1 $ and $ \Omega_2 $, respectively. (B) draws the time evolutions of $ l_1(t) $ and $ l_2(t) $. It is observed that $ l_2(t) $ is monotone increasing in time while $ l_1(t) $ goes toward $ O $ for a while and then turns apart from $ O $. (C) is the enlargement of the part surrounded by the dot circle in (B). The movement of $ {^t}(l_1(t), l_2(t)) $ approaching the set $ \tilde {m}_2 $ is also observed

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Figure 5. Movement of two front solutions for (23) in the case when $ R = 4 $ and $ r = 2 $ which corresponds to the case of 2-b). (A) denotes the locations $ l_1(t) $, $ l_2(t) $ of the zero level sets of the front solutions on $ \Omega_1 $ and $ \Omega_2 $, respectively. (B) draws the time evolutions of $ l_1(t) $ and $ l_2(t) $. It is observed that $ {^t}(l_1(t), l_2(t)) $ approaches a stationary state in the set $ \tilde {m}_2 $

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Figure 6. Movement of two front solutions for (23) corresponding to 2-c) when $ l_2(0) - l_1(0) > \frac{\log 2}{\alpha} $ is satisfied. (A) denotes the locations $ l_1(t) $, $ l_2(t) $ of the zero level points of the front solutions on $ \Omega_1 $ and $ \Omega_2 $, respectively. (B) draws the time evolutions of $ l_1(t) $ and $ l_2(t) $. It is observed that $ l_2(t) $ is monotone decreasing in time while $ l_1(t) $ goes apart from $ O $ for a while and then turns toward $ O $. The movement of $ {^t}(l_1(t), l_2(t)) $ approaching the set $ \tilde {m}_2 $ is also observed

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Figure 7. The profile of a stationary pulse solution of (24)

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