Electron. J. Diff. Eqns., Vol. 2005(2005), No. 54, pp. 1-22.

Local stability of spike steady states in a simplified Gierer-Meinhardt system

Georgi E. Karadzhov, David Edmunds, Pieter de Groen

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 $D$ less thatn $D(\varepsilon)$. Thus we extend the previous results known only for the case $r=p+1$ or $r=2, 1$ less than $p$ less than $5$.

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.

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Georgi E. Karadzhov
Institute for Mathematics and Informatics
Bulgarian academy of Sciences
1113 Sofia, Bulgaria
email: geremika@math.bas.bg
David E. Edmunds
Department of Mathematics
University of Sussex
Brighton BN1 9RF, U. K.
email: d.e.edmunds@sussex.ac.uk
Pieter P. N. de Groen
Department of Mathematics
Vrije Universiteit Brussel
Pleinlaan 2, B-1050 Brussels, Belgium
email: pdegroen@vub.ac.be

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