AN INSTABILITY CRITERION FOR ACTIVATOR INHIBITOR

Abstract. Let B be a two-dimensional ball with radius R. Let (u(x, y), ξ) be a non-constant steady state of the shadow system ut = Du∆u+ f(u, ξ) in B×R+ and 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 FitzHugh-Nagumo system. We show that if Z [Uθ ( ·)] ≥ 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 ( · ) ∈ C0(R/2piZ). The contrapositive of this result is the following: if (u, ξ) is stable for some τ> 0, then Z [Uθ ( ·)] = 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