Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 54, pp. 1-22.
Title: Local stability of spike steady states in a simplified
Gierer-Meinhardt system
Authors: Georgi E. Karadzhov (Bulgarian Academy of Sciences, bulgaria)
David Edmunds (Univ. of Sussex, Brighton BN1 9RF, U. K.)
Pieter de Groen (Vrije Univ. Brussel, Belgium)
Abstract:
In this paper we study the stability of the single internal spike
solution of a simplified Gierer-Meinhardt' system of equations
in one space dimension.
The linearization around this spike consists of a selfadjoint
differential operator plus a non-local term, which is a non-selfadjoint
compact integral operator.
We find the asymptotic behaviour of the small eigenvalues and we prove
stability of the steady state for the parameter $(p,q,r,\mu)$ in a
four-dimensional region (the same as for the shadow equation,
[8]) and for any finite $D$ if $\varepsilon$ is sufficiently small.
Moreover, there exists an exponentially large $D(\varepsilon)$ such that
the stability is still valid for $D