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

Abstract

Let B be a two-dimensional ball with radius R. Let (u(x, y), ξ ) be a nonconstant steady state of the shadow systemwhere f and g satisfy the following: f ξ (u, ξ ) < 0, g ξ (u, ξ ) < 0 and there is a function k(ξ ) such that g u (u, ξ ) = k(ξ )f ξ (u, ξ ). This system includes a special case of the Gierer-Meinhardt system and the FitzHugh-Nagumo system. We show that if Z[U θ (·)] 3, then (u, ξ ) is unstable for all τ > 0, where U(θ) := u(R cos θ, R sin θ) and Z[w(·)] denotes the cardinal number of the zero level set of w(·) ∈ C 0 (R/2π Z). The contrapositive of this result is the following: if (u, ξ ) is stable for some τ > 0, then Z[U θ (·)] = 2. In the proof of these results, we use a strong continuation property of partial differential operators of second order on the boundary of the domain.

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

“…Proposition 2.1 was obtained by [Mi06,Y02c]. However, for the completeness of the paper we prove Proposition 2.1.…”

“…Proposition 2.7 (Lemma 4.3 of [Mi06]). Let Ω(⊂ R 2 ) be a bounded domain with piecewise C 2 boundary, and let V ∈ C 0 (Ω).…”

“…Moreover, this stable boundary one-spike layer satisfies (1.2) [LT01,L01]. From the result of [Mi06] we cannot obtain the information of the shape of u in the whole domain. However, the necessary condition (1.2) and the existence of the stable boundary one-spike layer of the Gierer-Meinhardt system seem to suggest that the shape of the stable steady states of (SS B r 1 ) is like a boundary one-spike layer.…”

“…However, the shape of the stable steady states of a large class of reaction-diffusion systems in high-dimensional domains seems not to be known very much. In [Mi06], the author has given a necessary condition about the shape of the stable steady states of (SS B r 1 ) on the boundary, when the domain is a two-dimensional ball. Specifically, if (u, ξ) is stable for some τ > 0, then u is constant or…”

Point-condensation phenomena and saturation effect for the one-dimensional Gierer–Meinhardt system

“…In the case κ = 0 and N = 2, multi-interior peak stationary solutions were constructed and the stability was discussed in [19][20][21]. With respect to the stability analysis for the Gierer-Meinhardt system and its shadow system, see [12,7,9], and the references therein. Some a priori estimate for a stationary solutions to (2) were given in [6,2,13].…”