Weak Inner Layer in the Reaction-Diffusion-Advection Problem in the Case of a Reaction Discontinuity

The present work is devoted to the study of a one-dimensional reaction-advection-diffusion equation with weak smooth advection and a discontinuous reaction in the spatial coordinate. The work includes the construction of the asymptotics, proof of existence, and investigation of the stability of stationary solutions with the constructed asymptotics, which exhibit a weak inner layer that forms near the discontinuity point. For constructing the asymptotics, the method of A.B. Vasilieva was used; for proving the existence of the solution, the asymptotic method of differential inequalities was employed; and for studying the stability, the method of contracting barriers was applied. It is shown that such a solution, as the solution of the corresponding initial-boundary value problem, is asymptotically stable in the sense of Lyapunov. The stability region of finite (not asymptotically small) width for such a solution is specified, and it is established that the stationary problem has a unique solution in this region.