Upper and lower solutions for the FitzHugh–Nagumo type system of equations

We consider a moving front solution of a singularly perturbed FitzHugh–Nagumo type system of equations. The solution contains an internal transition layer, that is, a subdomain where a sharp change in the values of the functions describing the solution occurs. In initial-boundary value problems with moving front solutions, there naturally exists a small parameter that is equal to the ratio of the inner transition layer width to the width of the considered region. Taking into account this small parameter leads to the fact that the equations become singularly perturbed, thus the problems are classified as “hard”, the numerical solution of which meets certain difficulties and does not always give a reliable result. In connection with this, the role of an analytical investigation of the existence of a solution with an internal transition layer increases. For these purposes the use of differential inequalities method is especially effective. The method consists in constructing continuous functions, which are called upper and lower solutions. An important role is played by the so-called ”quasimonotonicity condition” for functions which describe reactive terms. In this paper, we present an algorithm for constructing the upper and the lower solutions of a parabolic system with a single-scale internal transition layer. It should be mentioned that the quasimonotonicity condition in the present paper differs from the analogous condition in previous publications. The above algorithm can be further generalized to more complex systems with two-scale transition layers or to systems with discontinuous reactive terms. The study is of great practical importance for creating mathematically grounded models in biophysics.