On a Variational Problem Arising from the Three-component FitzHugh-Nagumo Type Reaction-Diffusion Systems

Abstract

We study a variational problem arising from the three-component Fitzhugh-Nagumo type reaction diffusion systems and its shadow systems. In [15], Oshita studied the two-component systems. He revealed that a minimizer of energy corresponding to the problem oscillates under an appropriate condition and also studied its stability. Moreover, he mentioned its energy estimate without a proof. We investigate the behavior of a minimizer corresponding to the three-component problem, its stability and its energy estimate and extend some results of Oshita to the three-component systems and its shadow systems. In particular, we give a necessary and sufficient condition that the minimizer highly oscillates as $ \epsilon \to 0 $. Also, we establish a precise order of the energy estimate of the minimizer as $ \epsilon \to 0 $. In the proof of the energy estimate, we propose a new interpolation inequality.