The Gierer-Meinhardt system on a compact two-dimensional Riemannian manifold: Interaction of Gaussian curvature and Green's function
Let \((S,g)\) be a compact Riemann surface without boundary. The authors look for non-trivial steady states to the Gierer-Meinhardt system (which is used to model morphogenesis in biology) on \((S,g)\) and study the weak coupling case. Various authors in the past have shown there are point condensation phenomena -- i.e., the peaks become narrow and shrink to a set of points. In the current paper, the authors consider a single spike solution on \((S,g)\) using the Lyapunov-Schmidt reduction. A convex combination of the Gaussian curvature and Green's function (and their derivatives) is linked to the peak locations and the \(o(1)\) eigenvalues. A non local eigenvalue problem determines the \(O(1)\) eigenvalues which have negative part in this case.
scientific article; zbMATH DE number 5802610
scientific article