The Existence of a Boundary-Layer Stationary Solution to a Reaction–Diffusion Equation with Singularly Perturbed Neumann Boundary Condition

Abstract: This paper considers an initial-boundary value problem for a reaction–diffusion equation with a singularly perturbed Neumann boundary condition in a closed, simply connected two-dimensional domain. From a physical point of view, the problem describes processes with an intensive flow through the boundary of a given area. The existence of a stationary solution is proved, its asymptotic is constructed, and the Lyapunov stability conditions for it are established. The asymptotics of the solution are constructed by the classical Vasilieva algorithm using the Lusternik–Vishik method. The existence and stability of the solution are proved using the asymptotic method of differential inequalities.