On boundary layer solutions to a reaction-diffusion problem with a singularly perturbed nonlinear integral-type boundary value condition

The talk considers a boundary value problem for a nonlinear singularly perturbed reaction-diffusion equation with a nonlinear singularly perturbed integral-type boundary condition. The main result is a modification of Vasil'eva's boundary value function method for constructing the asymptotics of solutions with a boundary layer, the development of an asymptotic method of differential inequalities to substantiate the existence of solutions uniformly close to the constructed asymptotics in the domain under consideration, and a proof of the asymptotic stability in the sense of Lyapunov of such solutions as solutions of the corresponding initial-boundary value problem. These results are obtained for two cases: either the presence or absence of quasi-monotonicity in the integrand. Examples illustrating the results are given.