Pinned solutions in a heterogeneous three-component FitzHugh-Nagumo model. (English) Zbl 1415.35018
This article presents an analysis of a singularly perturbed three-component FitzHugh-Nagumo model with a small jump-type heterogeneity. The authors consider pinned front solutions and pulse solutions for the model problem. Explicit conditions for the existence and stability of the pinned front solutions are demonstrated. For this, geometric singular perturbation techniques with an action functional approach are employed (the interested reader should also see [A. Doelman et al., SIAM J. Appl. Dyn. Syst. 15, No. 2, 655–712 (2016; Zbl 1343.34041)]). One novelty in this article is the determination of particular parameter ranges where one can explicitly compute the pinning distance of a local defect solution. It should also be mentioned that local defect solutions were investigated numerically in [P. van Heijster et al., Nonlinearity 24, No. 1, 127–157 (2011; Zbl 1208.35010)].
Reviewer: Joseph Shomberg (Providence)
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 35K57 | Reaction-diffusion equations |
| 49J40 | Variational inequalities |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 34A36 | Discontinuous ordinary differential equations |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
Keywords:
localised defect solutions; small jump-type heterogeneity; pinned front solutions; pulse solutions; pinning distanceReferences:
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