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|Title:||The Gierer-Meinhardt system on a compact two-dimensional Riemannian Manifold: Interaction of Gaussian curvature and Green's function|
|Keywords:||Pattern formation;Mathematical biology;Singular perturbation;Riemannian manifold|
|Citation:||Journal de Mathematiques Pures et Appliquees. 94(4): 366–397, Oct 2010|
|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 is small enough. We show that for a threshold ratio of the activator diffusivity and the inhibitor diffusivity, 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.|
|Appears in Collections:||Dept of Mathematics Research Papers|
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