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RETICAL AND MATHEMATICAL PHYSICS

The Existence and Stability of Periodic Solutions with a Boundary Layer in a Two-Dimensional Reaction-Diffusion Problem in the Case of Singularly Perturbed Boundary Conditions of the Second Kind N. N. Nefedov1* and E. I. Nikulin1** 1

Department of Physics, Moscow State University, Moscow, 119991 Russia Received October 29, 2019; revised December 10, 2019; accepted December 17, 2019

Abstract—The existence of time-periodic solutions of the boundary-layer type to a two-dimensional reaction–diffusion problem with a small-parameter coefficient of a parabolic operator is proved in the case of singularly perturbed boundary conditions of the second kind. An asymptotic approximation with respect to the small parameter is constructed for these solutions. The set of boundary conditions for which these solutions exist is studied and the local uniqueness and asymptotic Lyapunov stability are established for them. It is shown that, unlike the analogous Dirichlet problem, for which such a solution is unique, there can be several solutions of this kind for the problem under consideration, each of which has its domains of stability and local uniqueness. To prove these facts, results based on the asymptotic principle of differential inequalities are used. Keywords: singularly perturbed parabolic problems, periodic problems, reaction-diffusion equations, boundary layers, asymptotic methods, differential inequalities, asymptotic Lyapunov stability. DOI: 10.3103/S0027134920020083

INTRODUCTION Periodic parabolic boundary value problems have been being intensively studied from both the theoretical and applied standpoints. A number of problem classes that are important for applications can be found, for example, in [1]. Evolution and periodic problems with analogous higher differential operators were considered in [3–6] based on the operator method under development, as well as their application in a number of applied physical problems. In applications, equations of this kind are called reactiondiffusion equations or reaction–diffusion–advection equations when they involve a term that describes transport. These equations are widely and successfully used in nonlinear wave theory and fluid dynamics (see, for example, [2–8]). In many cases, singularly perturbed problems of the considered type are used to describe processes with an intense reaction (source), namely, problems with a small-parameter coefficient of the higher differential operator. A characteristic feature of these problems is the existence of solutions with boundary and inner transition layers [9–11]. We also note that an analogous method was applied *

E-mail: [email protected] ** E-mail: [email protected]

in [12] to study the asymptotic stability. Reaction– diffusion–advection equations often occur in applications, for example, in ecology when the variation in temperature or gas concentration in surface layers of the atmosphere is mathematically simulated, as well as in chemical kinetics and biological kinetics. In this work, we consider a new clas