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

Shin-Ichiro Ei | Zendy; K. Ikeda | Zendy; Yasuhito Miyamoto | Zendy

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

Author(s) -

Shin-Ichiro Ei,

K. Ikeda,

Yasuhito Miyamoto

Publication year - 2011

Publication title -

communications on pure andamp applied analysis

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.077

H-Index - 42

eISSN - 1553-5258

pISSN - 1534-0392

DOI - 10.3934/cpaa.2012.11.115

Subject(s) - boundary (topology) , curvature , shadow (psychology) , spike (software development) , boundary value problem , mathematical analysis , physics , pattern formation , mathematics , geometry , computer science , psychology , software engineering , biology , psychotherapist , genetics

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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