Spike solutions to a nonlocal differential equation (Q2469350)

This paper deals with the one-dimensional reaction-diffusion system: \[ u_t=\varepsilon^{2}u_{xx}-u+ \frac{u^{\alpha}}{v^{\beta}},\quad-\infty<x<+\infty,\;t>0, \] \[ v=\left(\frac{\varepsilon^{-1}}{2\mu}\int_{-\infty}^{+\infty} u^{\gamma}(x,t)\,dx\right)^{\frac{1}{\theta+1}},\quad t>0 \] \[ 0<u(x,t)\to 0 \quad \text{as } x\to \infty, \] which comes from a Gierer-Meinhardt model, on a finite interval, when the inhibitor diffusivity \(D\to \infty\), the activator diffusivity \(d\to 0\) and the reaction time constant \(\tau=0\). All constants are positive and satisfy the condition \(0<\frac{\alpha-1}{\beta}< \frac{\gamma}{\theta+1}\). The authors show the existence of a unique spike steady state solution with the only spike located at \(x=0\) and study its stability. Under the conditions \(\gamma=2\), \(1<\alpha\leq 5\) or \(\gamma= \alpha+1\), \(\alpha>1\) and thanks to a lemma of \textit{D. Iron, M. Ward}, and \textit{J. Wei} [Physica D 150, No. 1--2, 25--62 (2001; Zbl 0983.35020)], they prove, in particular, that the real part of the eigenvalues of the associated linearized eigenvalue problem are non-positive, and they present an asymptotic analysis of slow spike motions as \(\varepsilon\to 0\).