# American Institute of Mathematical Sciences

May  2018, 38(5): 2441-2465. doi: 10.3934/dcds.2018101

## The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity

 Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan

Received  August 2017 Published  March 2018

We construct a heteroclinic solution to the FitzHugh-Nagumo type reaction-diffusion system (FHN RD system) with heterogeneity by the sub-supersolution method due to [5]. $σ(d,γ)$ is introduced as the Rayleigh quotient corresponding to a linearized eigenvalue problem of the subsolution, where $d$ and $γ$ are parameters. The key to construct the solution is the uniform estimate for $σ(·,·)$ from below. In addition, it enables us to analyze an asymptotic behavior of the solution.

Citation: Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
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