Electron. J. Diff. Eqns., Vol. 2005(2005), No. 54, pp. 122.
Local stability of spike steady states in a simplified
GiererMeinhardt system
Georgi E. Karadzhov, David Edmunds, Pieter de Groen
Abstract:
In this paper we study the stability of the single internal spike
solution of a simplified GiererMeinhardt' system of equations
in one space dimension.
The linearization around this spike consists of a selfadjoint
differential operator plus a nonlocal term, which is a nonselfadjoint
compact integral operator.
We find the asymptotic behaviour of the small eigenvalues and we prove
stability of the steady state for the parameter
in a
fourdimensional region (the same as for the shadow equation,
[8]) and for any finite
if
is sufficiently small.
Moreover, there exists an exponentially large
such that
the stability is still valid for
. Thus we extend the
previous results known only for the case
or
.
Submitted March 3, 2005. Published May 23, 2005.
Math Subject Classifications: 35B25, 35K60.
Key Words: Spike solution; singular perturbations;
reactiondiffusion equations; GiererMeinhardt equations.
Show me the
PDF file (319K),
TEX file, and other files for this article.

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, B1050 Brussels, Belgium
email: pdegroen@vub.ac.be 
Return to the EJDE web page