Internal Layer for a Singularly Perturbed Equation with Discontinuous Right-Hand Side

Abstract: We consider a boundary value problem for an ordinary singularly perturbed second-orderdifferential equation whose right-hand side is a nonlinear function with a discontinuity along somecurve that is independent of the small parameter. For this problem, we study the existence ofa smooth solution with steep gradient in a neighborhood of some point lying on this curve. Thepoint itself and an asymptotic representation for the solution are to be determined. The existencetheorem is proved by the method of matching asymptotic expansions. To this end, we usetheorems on existence of solutions of boundary value problems for singularly perturbed equationsand methods for constructing asymptotic approximations to these solutions.