Local stability of spike steady states in a simplified Gierer-Meinhardt system

Abstract

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, μ) in a four-dimensional region (the same as for the shadow equation, [8]) and for any finite D if ε is sufficiently small. Moreover, there exists an exponentially large D(ε) such that the stability is still valid for D < D(ε). Thus we extend the previous results known only for the case r = p + 1 or r = 2, 1 < p < 5.