Boundary Layer Lines of Solutions to Singularly Perturbate Equations With a Saddle Point

Abstract

In this paper, we study the asymptotic behavior of solutions of singularly perturbed equations in complex domains. The considered equations have a saddle point. The main goal is to prove the existence of boundary-layer lines, boundary-layer regions and regular, singular regions, identifying their features in comparison with previous studies. Harmonic functions and their level lines are involved in solving the set problem. Using the level line, geometric constructions are carried out. The area under consideration is divided into parts and integration paths are chosen that ensure the convergence of some functions with respect to a small parameter. Using the method of successive approximations, the existence and boundedness of the solution of the equation is proved. The features of the boundary lines are revealed.