open access

https://hdl.handle.net/2115/5419

Stability of a boundary spike layer for the Gierer–Meinhardt system

英語

2005-08-23

Cambridge University Press

European Journal of Applied Mathematics

16(4)

467-491

We consider the activator-inhibitor Gierer–Meinhardt reaction-diffusion system of biological pattern formation in a closed bounded domain. The existence and stability of a boundary apike-layer solution to the Gierer–Meinhardt model, and it, so-called shadow limit, is analysed. In the limit of small activator diffusivity, together with a large inhibitor diffusivity, an equilibrium boundary spike-layer solution is constructed that concentrates at a non-degenerate critical point $P$ of the boundary. By non-degenerate we mean that every principal curvature of the boundary has a local maximum at $P$, and hence the mean curvature at the boundary has a local maximum at $P$. Rigorous results for the stability of such a boundary spike-layer solution are given.

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