The Gierer–Meinhardt system on a compact two-dimensional Riemannian manifold: Interaction of Gaussian curvature and Green's function

Abstract

In this paper, we rigorously prove the existence and stability of single-peaked patterns for the singularly perturbed Gierer–Meinhardt system on a compact two-dimensional Riemannian manifold without boundary which are far from spatial homogeneity. Throughout the paper we assume that the activator diffusivity ϵ 2 is small enough. We show that for the threshold ratio D ∼ 1 ϵ 2 of the activator diffusivity ϵ 2 and the inhibitor diffusivity D, the Gaussian curvature and the Green's function interact. A convex combination of the Gaussian curvature and the Green's function together with their derivatives are linked to the peak locations and the o ( 1 ) eigenvalues. A nonlocal eigenvalue problem (NLEP) determines the O ( 1 ) eigenvalues which all have negative part in this case.