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 u(t) = D-u Delta u + f(u,xi) in B x R+ and tau xi(t) = 1/vertical bar B vertical bar integral(B)integral g(u,xi) dx dy in R+, partial derivative(nu)u = 0 on partial derivative B x R+, where f and g satisfy the following: f xi(u,xi) < 0, g xi(u,xi) < 0, and there is a function k(xi) such that g(u)(u,xi) = k(xi) f xi (u,xi). 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,xi) is stable for some tau > 0, then the maximum (minimum) of u is attained at exactly one point on a B and u has no critical point in B \ partial derivative B. In proving this result, we prove a nonlinear version of the "hot spots" conjecture of J. Rauch in the case of B. (c) 2007 Elsevier Inc. All rights reserved.