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

Let B be a two-dimensional ball with radius R. We continue to study the shape of the stable steady states tout=DuΔu+f(u,ξ)in B×R+andτξt=1|B|∫∫Bg(u,ξ)dxdyin R+,∂νu=0on ∂B×R+, where f and g   satisfy the following: fξ(u,ξ)<0fξ(u,ξ)<0, gξ(u,ξ)<0gξ(u,ξ)<0, and there is a function k(ξ)k(ξ) such that gu(u,ξ)=k(ξ)fξ(u,ξ)gu(u,ξ)=k(ξ)fξ(u,ξ). This system includes a special case of the Gierer–Meinhardt system and the shadow system with the FitzHugh–Nagumo type nonlinearity. We show that, if the steady state (u,ξ)(u,ξ) is stable for some τ>0τ>0, then the maximum (minimum) of u is attained at exactly one point on ∂B and u   has no critical point in B∖∂BB∖∂B. In proving this result, we prove a nonlinear version of the “hot spots” conjecture of J. Rauch in the case of B.