On a weak periodic internal layer in a problem with a discontinuous reaction

E. I. Nikulin, A. V. Karamyshev

Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia

Abstract: We consider a boundary value problem with a time-periodic condition for an equation of “reaction-advection-diffusion” type with weak smooth advection and with reaction discontinuous in the spatial coordinate. We construct the asymptotics, prove the existence, and investigate the stability of periodic solutions with the constructed asymptotics and with a weak internal layer formed near the discontinuity point. To construct the asymptotics, we use the A. B. Vasil'eva method; to justify the existence of the solution, the asymptotic method of differential inequalities; and to study stability, the method of contracting barriers. We show that such a solution, as a solution to the corresponding initial-boundary value problem, is asymptotically Lyapunov stable. We determine the stability domain of a finite (not asymptotically small) width for such a solution and prove that the solution to the periodic problem is unique in this domain.

Keywords: periodic problem, singularly perturbed equations, reaction-advection-diffusion equations, internal layer, asymptotic methods, method of differential inequalities, asymptotic Lyapunov stability, drift-diffusion semiconductor model.

PACS: 02.30.Jr., 72.20.-i.

Received: 01.03.2025
Revised: 26.04.2025

DOI: 10.4213/tmf10961

 English version: Theoretical and Mathematical Physics, 2025, 224:2, 1414–1427

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