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 $D<D(\varepsilon)$. Thus we extend the
 previous results known only for the case $r=p+1$ or $r=2, 1<p<5$.
 
Submitted March 3, 2005. Published May 23, 2005.
Math Subject Classifications: 35B25, 35K60.
Key Words: Spike solution; singular perturbations;
           reaction-diffusion equations; Gierer-Meinhardt equations.