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

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 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 of 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 of the periodic problem is unique in this domain.