An asymptotic reduction of the Gierer–Meinhardt activator-inhibitor system in the limit of
large inhibitor diffusivity and small activator diffusivity ε leads to a singularly perturbed
nonlocal reaction-diffusion equation for the activator concentration. In the limit ε → 0, this
nonlocal problem for the activator concentration has localized spike-type solutions. In this
limit, we analyze the motion of a spike that is confined to the smooth boundary of a two or
three-dimensional domain. By deriving asymptotic differential equations for the spike motion,
it is shown that the spike moves towards a local maximum of the curvature in two dimensions
and a local maximum of the mean curvature in three dimensions. The motion of a spike
on a flat segment of a two-dimensional domain is also analyzed, and this motion is found
to be metastable. The critical feature that allows for the slow boundary spike motion is the
presence of the nonlocal term in the underlying reaction-diffusion equation.
Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation
