Global Solutions and Long-Time Behavior to a Fitzhugh–Nagumo Myelinated Axon Model

Abstract

A class of problems of the following form is considered: \[ \begin{gathered} u_t = u_{xx} - gu\quad x \in \mathbb{R}\backslash Z, \hfill u_t = u_x (n + ,t) - u_x (n - ,t) + f(u) - w\quad x = n \in Z, \hfill w_t = \sigma u - \gamma w\quad x = n \in Z, \hfill \end{gathered} \] In particular, the existence of solutions to the Cauchy problem is proved and the time evolution of solutions to the problem is studied. Such a system models an infinite, myelinated axon with discrete, excitable nodes spaced unit distance apart, and the model dynamics are of Fitzhugh–Nagumo type.