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 to ut = Du∆u + f(u, ξ) in B× R+, τξt = 1 |B| ZZ B g(u, ξ)dxdy in R+, ∂νu = 0 on ∂B × R+, 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 shadow system with the FitzHugh-Nagumo type nonlinearity. We show that, if the steady state (u, ξ) is stable for some τ > 0, then the maximum (minimum) of u is attained at exactly one point on ∂B and u has no critical point in B\∂B. In proving this results, we prove a nonlinear version of the “hot spots” conjecture of J. Rauch in the case of B.