Electron. J. Diff. Eqns., Vol. 2002(2002), No. 50, pp. 1-22.

Metastability in the shadow system for Gierer-Meinhardt's equations

Pieter de Groen & Georgi Karadzhov

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

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Pieter de Groen
Vrije Universiteit Brussel
Pleinlaan 2
B-1050 Brussels, Belgium
email: pdegroen@vub.ac.be
Georgi E. Karadzhov
Bulgarian academy of Sciences
Institute for Mathematics and Informatics
Sofia, Bulgaria
email: geremika@math.bas.bg

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