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 diusivity 2 is small enough. We show that for the threshold ratio D ∼
Existence and stability of a single spike solution for the Gierer-Meinhardt system on a Riemannian manifold has been shown in REF .
THE GIERER-MEINHARDT SYSTEM ON A COMPACT TWO-DIMENSIONAL RIEMANNIAN MANIFOLD: INTERACTION OF GAUSSIAN CURVATURE AND GREEN'S FUNCTION
Journal de Mathématiques Pures et Appliquées
Mathematics