Geodesic
Metastability in the shadow system for Gierer-Meinhardt's equations
Electronic journal of differential equations, Tome 2002 (2002)
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__a197,
     author = {De Groen,  Pieter and Karadzhov,  Georgi},
     title = {Metastability in the shadow system for {Gierer-Meinhardt's} equations},
     journal = {Electronic journal of differential equations},
     year = {2002},
     volume = {2002},
     zbl = {1002.35014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a197/}
}
TY  - JOUR
AU  - De Groen,  Pieter
AU  - Karadzhov,  Georgi
TI  - Metastability in the shadow system for Gierer-Meinhardt's equations
JO  - Electronic journal of differential equations
PY  - 2002
VL  - 2002
UR  - http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a197/
LA  - en
ID  - EJDE_2002__2002__a197
ER  - 
%0 Journal Article
%A De Groen,  Pieter
%A Karadzhov,  Georgi
%T Metastability in the shadow system for Gierer-Meinhardt's equations
%J Electronic journal of differential equations
%D 2002
%V 2002
%U http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a197/
%G en
%F EJDE_2002__2002__a197
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__a197/