Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model

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In the limit of small activator diffusivity $\varepsilon$, and in a bounded domain in $\mathbb{R}^{N}$ with $N=1$ or $N=2$ under homogeneous Neumann boundary conditions, the bifurcation behavior of an equilibrium one-spike solution to the Gierer-Meinhardt activator-inhibitor system is analyzed for different ranges of the inhibitor diffusivity $D$. When $D=\infty$, it is well-known that a one-spike solution for the resulting shadow Gierer-Meinhardt system is unstable, and the locations of unstable equilibria coincide with the points in the domain that are furthest away from the boundary. For a unit disk domain it is shown that as $D$ is decreased below a critical bifurcation value $D_{c}$, with $D_{c}=O(\varepsilon^2 e^{2/\varepsilon})$, the spike at the origin becomes stable, and unstable spike solutions bifurcate from the origin. The locations of these bifurcating spikes tend to the boundary of the domain as $D$ is decreased further. Similar bifurcation behavior is studied in a one-parameter family of dumbbell-shaped domains. This motivates a further analysis of the existence of certain near-boundary spikes. Their location and stability is given in terms of the modified Green's function. Finally, for the dumbbell-shaped domain, an intricate bifurcation structure is observed numerically as $D$ is decreased below some $O(1)$ critical value.