Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case. (English) Zbl 1141.35345

Wei, J.; Winter, M.

J. Nonlinear Sci. 11, No. 6, 415-458 (2001).

Summary: We rigorously prove the existence and stability of multiple-peaked patterns that are far from spatial homogeneity for the singularly perturbed Gierer-Meinhardt system in a two-dimensional domain. 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 instability; if it lies below, then we get stability.

MSC:

35B45	A priori estimates in context of PDEs
35J40	Boundary value problems for higher-order elliptic equations

Keywords:

Pattern formation; mathematical biology; singular perturbation; weak coupling