Asymptotics of a Spike Type Contrast Structure in a Problem with a Multiple Root of the Degenerate Equation

Abstract

We consider a boundary value problem for a singularly perturbed second-order ordinary differential equation for the case in which the corresponding degenerate equation has a double root. We prove that under some conditions the problem has a solution that is close to this root everywhere except for a small neighborhood of some point, where this solution exhibits a spike. This neighborhood consists of several zones differing in the nature of rapid change in the solution; this is explained by the fact that the root of the degenerate equation is multiple. The asymptotics is constructed and justified for this solution, which is called a spike type contrast structure.