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

Let B be a two-dimensional ball with radius R. Let (u(x,y),ξ) be a nonconstant steady state of the shadow system where f and g satisfy the following: fξ(u,ξ)<0, gξ(u,ξ)<0 and there is a function k(ξ) such that gu(u,ξ)=k(ξ)fξ(u,ξ). This system includes a special case of the Gierer–Meinhardt system and the FitzHugh–Nagumo system. We show that if , then (u,ξ) is unstable for all τ>0, where U(θ):=u(Rcosθ,Rsinθ) and denotes the cardinal number of the zero level set of . The contrapositive of this result is the following: if (u,ξ) is stable for some τ>0, then . 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.