Geodesic
Local stability of spike steady states in a simplified Gierer-Meinhardt system
Electronic journal of differential equations, Tome 2005 (2005)
In this paper we study the stability of the single internal spike solution of a simplified Gierer-Meinhardt' system of equations in one space dimension. The linearization around this spike consists of a selfadjoint differential operator plus a non-local term, which is a non-selfadjoint compact integral operator. We find the asymptotic behaviour of the small eigenvalues and we prove stability of the steady state for the parameter $(p,q,r,\mu)$ in a four-dimensional region (the same as for the shadow equation, [8]) and for any finite $D$ if $\varepsilon$ is sufficiently small. Moreover, there exists an exponentially large $D(\varepsilon)$ such that the stability is still valid for $r=p+1$ or .
Classification : 35B25, 35K60
Keywords: spike solution, singular perturbations, reaction-diffusion equations, Gierer-Meinhardt equations 
@article{EJDE_2005__2005__a229,
     author = {Karadzhov,  Georgi E. and Edmunds,  David and de Groen,  Pieter},
     title = {Local stability of spike steady states in a simplified {Gierer-Meinhardt} system},
     journal = {Electronic journal of differential equations},
     year = {2005},
     volume = {2005},
     zbl = {1077.35055},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a229/}
}
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AU  - de Groen,  Pieter
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%A de Groen,  Pieter
%T Local stability of spike steady states in a simplified Gierer-Meinhardt system
%J Electronic journal of differential equations
%D 2005
%V 2005
%U http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a229/
%G en
%F EJDE_2005__2005__a229
Karadzhov,  Georgi E.; Edmunds,  David; de Groen,  Pieter. Local stability of spike steady states in a simplified Gierer-Meinhardt system. Electronic journal of differential equations, Tome 2005 (2005). http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a229/