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June 2018 On a Variational Problem Arising from the Three-component FitzHugh-Nagumo Type Reaction-Diffusion Systems
Takashi KAJIWARA, Kazuhiro KURATA
Tokyo J. Math. 41(1): 131-174 (June 2018). DOI: 10.3836/tjm/1502179257

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.

Citation

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Takashi KAJIWARA. Kazuhiro KURATA. "On a Variational Problem Arising from the Three-component FitzHugh-Nagumo Type Reaction-Diffusion Systems." Tokyo J. Math. 41 (1) 131 - 174, June 2018. https://doi.org/10.3836/tjm/1502179257

Information

Published: June 2018
First available in Project Euclid: 18 December 2017

zbMATH: 06966862
MathSciNet: MR3830812
Digital Object Identifier: 10.3836/tjm/1502179257

Subjects:
Primary: 35J50
Secondary: 35K57, 35Q92

Rights: Copyright © 2018 Publication Committee for the Tokyo Journal of Mathematics

JOURNAL ARTICLE
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Vol.41 • No. 1 • June 2018
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