Spikes for the Two-Dimensional Gierer-Meinhardt System: The Weak Coupling Case

Abstract

In this paper, we rigorously prove the existence and stability of multiple-peaked patterns for the singularly perturbed Gierer-Meinhardt system in a two dimensional domain which are far from spatial homogeneity. The Green's function together with its derivatives is linked to the peak locations and to the $o(1)$ eigenvalues, which vanish in the limit. On the other hand two nonlocal eigenvalue problems (NLEPs), one of which is new, are related to the O(1) eigenvalues. Under some geometric condition on the peak locations, we establish a threshold behavior: If the inhibitor diffusivity exceeds the threshold then we get stability, if it lies below then we get instability.