Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 50, pp. 1-22.
Title: Metastability in the shadow system for Gierer-Meinhardt's equations
Authors: Pieter de Groen (Vrije Univ. Brussel, Belgium)
Georgi Karadzhov (Bulgarian academy of Sciences)
Abstract:
In this paper we study the stability of the single internal
spike solution of the shadow system for the
Gierer-Meinhardt equations in one space dimension.
It is well-known, that the linearization around this spike
consists of a differential operator plus a non-local term.
For parameter values in certain subsets of the 3D
$(p,q,r)$-parameter space we
prove that the non-local term moves the negative $O(1)$ eigenvalue
of the differential operator to the positive (stable) half plane
and that an exponentially small
eigenvalue remains in the negative half plane, indicating a
marginal instability (dubbed ``metastability''). We also
show, that for parameters $(p,q,r)$ in another region,
the $O(1)$ eigenvalue remains in the negative half plane.
In all asymptotic approximations we compute
rigorous bounds for the order of the error.
Submitted April 17, 2002. Published June 2, 2002.
Math Subject Classifications: 35B25, 35K60
Key Words: Spike solution; singular perturbations;
reaction-diffusion equations; Gierer-Meinhardt equations