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

Abstract

Abstract. We continue to study the shape of the stable steady states of the so-called shadow limit of activator-inhibitor systems in two-dimensional domainswhere f and g satisfy the following: g ξ < 0, and there is a function (u, ξ). This class of reaction-diffusion systems includes the FitzHughNagumo 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, ξ) 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, ξ) is stable for some τ > 0, then the shape of u is like a boundary one-spike layer even if D u is not small.

Dynamics and interactions of spikes on smoothly curved boundaries for reaction–diffusion systems in 2D

“…Thus, our results give the well-known results for a stable boundary spike solution of the Gierer-Meinhardt model [9,[11][12][13]16] from the viewpoint of dynamics. We can also show the existence of a stable stationary solution with two peaks in the neighborhood of a point with maximal curvature of ∂ by using the repulsive interaction (Fig.…”

“…Recently, [11] and [12] proved the existence and stability of a stationary spike solution with more than one peak at points with locally maximal mean curvatures of ∂ under the condition a 3 = a 1 + 1 in (1.1). Thus, there has been much research and many results on stationary spike solutions with peaks on boundaries for (1.1), but we do not know a result on the dynamics of spikes along boundaries.…”

An instability criterion for activator–inhibitor systems in a two-dimensional ball II

“…However, we cannot obtain information about u in the interior of the domain. One of the main results of author's previous paper [13,Theorem 4.7] is a partial answer of this question. In [13], we show that, if sup (ρ 1 ,ρ 2 )∈R 2 f u (ρ 1 , ρ 2 ) < D u κ 4 , then the conclusion of Theorem A below holds.…”

Dynamics of a boundary spike for the shadow Gierer-Meinhardt system

“…One of the authors showed in [18] that if r = p + 1 and τ is sufficiently small, a stationary solution with a boundary spike layer near a non-degenerate local maximum point of the curvature of ∂Ω is stable. The problems of existence and stability of spikes have large literature.…”