Multizonal Internal Layers in a Stationary Piecewise–Smooth Reaction-Diffusion Equation in the Case of the Difference of Multiplicity for the Roots of the Degenerate Solution

A singularly perturbed stationary problem for a one-dimensional reaction-diffusion equation in the case when the degenerate equation has multiple roots is studied. This is a new class of problems with discontinuous reactive terms along some curve that is independent of the small parameter. The existence of a smooth solution with the transition from the triple root of one degenerate equation to the double root of the other degenerate equation in the neighborhood of some point on the discontinuous curve is studied. Based on the existence theorem of classical boundary value problems and the technique of matching asymptotic expansion, the existence of a smooth solution is proved. And the point itself and the asymptotic representation of this solution are constructed by the matching technique and modified boundary layer function method.