On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

We continue to study the shape of the stable steady states of the so-called shadow limit of activator-inhibitor systems in two-dimensional domains \begin{gather*} u_t=D_u\Delta u+f(u,\xi )\;\text {\ in\ }\; \Omega \times \mathbb {R}_+\text {\quad and\quad } \tau \xi _t=\frac {1}{|\Omega |}\iint _{\Omega }g(u,\xi )dxdy \;\text {\ in\ }\;\mathbb {R}_+,\\ \ \partial _{\nu }u=0\;\text {\ on\ }\; \partial \Omega \times \mathbb {R}_+, \end{gather*} where $f$ and $g$ satisfy the following: $g_{\xi }<0$, and there is a function $k(\xi )\in C^0$ such that $f_{\xi }(u,\xi )=k(\xi )g_{u}(u,\xi )$. This class of reaction-diffusion systems includes the FitzHugh-Nagumo system and a special case of the Gierer-Meinhardt system. In the author’s previous paper “An instability criterion for activator-inhibitor systems in a two-dimensional ball” (J. Diff. Eq. 229 (2006), 494–508), we obtain a necessary condition about the profile of $u$ on the boundary of the domain for a steady state $(u,\xi )$ to be stable when the domain is a two-dimensional ball. In this paper, we give a necessary condition about the profile of $u$ in the domain when the domain is a two-dimensional ball, annulus or rectangle. Roughly speaking, we show that if $(u,\xi )$ is stable for some $\tau >0$, then the shape of $u$ is like a boundary one-spike layer even if $D_u$ is not small.