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 $(p,q,r,mu)$ 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 $D$ less thatn $D(varepsilon)$. Thus we extend the previous results known only for the case $r=p+1$ or $r=2, 1$ less than $p$ less than $5$.

Keywords

• Spike solution
• singular perturbations
• reaction-diffusion equations
• Gierer-Meinhardt equations.