Geodesic
Metastability in the shadow system for Gierer-Meinhardt's equations
Electronic Journal of Differential Equations, Tome 2002 (2002).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 35B25, 35K60
Keywords: spike solution, singular perturbations, reaction-diffusion equations 
@article{EJDE_2002__2002__a297,
     author = {De Groen, Pieter and Karadzhov, Georgi},
     title = {Metastability in the shadow system for {Gierer-Meinhardt's} equations},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2002},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a297/}
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De Groen, Pieter; Karadzhov, Georgi. Metastability in the shadow system for Gierer-Meinhardt's equations. Electronic Journal of Differential Equations, Tome 2002 (2002). http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a297/