## A Singular Perturbation Problem for Heteroclinic Solutions to the FitzHugh-Nagumo Type Reaction-Diffusion System with Heterogeneity

J. Math. Sci. Univ. Tokyo
Vol. 26 (2019), No. 2, Page 141-199.

Kajiwara, Takashi; Kurata, Kazuhiro
A Singular Perturbation Problem for Heteroclinic Solutions to the FitzHugh-Nagumo Type Reaction-Diffusion System with Heterogeneity
In a previous paper, the first author considered the variational problems for heteroclinic solutions to the FitzHugh-Nagumo type reaction-diffusion system involving heterogeneity $\mu(x)$ and proved the existence of the minimizers. However, the precise location of the transition layer of the minimizers was not clear in the paper. In this paper, we consider the same problems as the singular perturbation problems. Then we prove that the minimizer has exactly one transition layer near the minimum point of $\mu(x)$ by using the first order energy expansion. Moreover, we derive the more precise energy asymptotic expansion.