Geodesic
Metastability in the shadow system for Gierer-Meinhardt's equations
Electronic journal of differential equations, Tome 2002 (2002)
Zbl   EuDML
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 
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/
@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/}
}
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