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 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
in a
four-dimensional 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;
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 |
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David E. Edmunds Department of Mathematics University of Sussex Brighton BN1 9RF, U. K. email: d.e.edmunds@sussex.ac.uk |
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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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