Dynamics of a boundary spike for the shadow Gierer-Meinhardt system

Shin-Ichiro Ei; Kota Ikeda; Yasuhito Miyamoto

doi:10.3934/cpaa.2012.11.115

Article Contents

2012, Volume 11, Issue 1: 115-145. Doi: 10.3934/cpaa.2012.11.115

This issue Previous Article Qualitative analysis and travelling wave solutions for the SI model with vertical transmission Next Article Approximation of nonlinear parabolic equations using a family of conformal and non-conformal schemes

Dynamics of a boundary spike for the shadow Gierer-Meinhardt system

1.
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395
2.
Graduate School of Advanced Mathematical Science, Meiji University, Kawasaki, 214-8571
3.
Department of Mathematics, Tokyo Institute of Technology, O-Okayama, Meguro-ku, Tokyo 152-8551

Received: November 30, 2009

Revised: September 30, 2010

Published: December 2011

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Abstract

The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. The authors of [3] showed that if an initial value is close to a spiky pattern and its peak is far away from the boundary, the solution of the shadow Gierer-Meinhardt system, called a interior spike solution, moves towards a point on boundary which is the closest to the peak. However it has not been studied how a solution close to a spiky pattern with the peak on the boundary, called a boundary spike solution moves along the boundary. In this paper, we consider the shadow Gierer-Meinhardt system and dynamics of a boundary spike solution. Our results state that a boundary spike moves towards a critical point of the curvature of the boundary and approaches a stable stationary solution.

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Mathematics Subject Classification: Primary 35B25, 35B35; Secondary 35K57, 35P15.

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