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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:239

Article

Miyamoto, Yasuhito

关键词: instability;    activator-inhibitor system;    shadow system;    reaction-diffusion systems;    nodal curve;    nodal domain;    second eigenvalue;   

DOI  :  10.1016/j.jde.2007.05.006

来源: Elsevier

【 摘 要 】

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.

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