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http://bura.brunel.ac.uk/handle/2438/4531
Title: | The Gierer-Meinhardt system on a compact two-dimensional Riemannian Manifold: Interaction of Gaussian curvature and Green's function |
Authors: | Tse, WH Wei, J Winter, M |
Keywords: | Pattern formation;Mathematical biology;Singular perturbation;Riemannian manifold |
Issue Date: | 2010 |
Publisher: | Elsevier |
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. |
URI: | http://bura.brunel.ac.uk/handle/2438/4531 |
DOI: | http://dx.doi.org/10.1016/j.matpur.2010.03.003 |
Appears in Collections: | Dept of Mathematics Research Papers Mathematical Sciences |
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