Mini-maximizers for reaction-diffusion systems with skew-gradient structure (Q1598380)

A reaction-diffusion system with skew-gradient structure consists of two gradient systems coupled in a skew-symmetric way, whose steady states correspond to critical points of a functional \(E(u,v)\). A mini-maximizer \((\varphi,\psi)\) of \(E(u,v)\) is a pair such that \(\varphi\) is a minimizer of \(E(u,\psi)\) and \(\psi\) a maximizer of \(E(\varphi,v)\). The author investigates relations between a stability property of steady states of the reaction-diffusion system and the mini-maximizing property for \(E(u,v)\). He proves in particular that a steady state of the system is stable regardless of time constants if and only if it is a mini-maximizer of the functional.