Existence and stability of the periodic solution with an interior transitional layer in the problem with a weak linear advection

In the paper, we study a singularly perturbed periodic in time problem for the parabolic reaction-advection-diffusion equation with a weak linear advection. The case of the reactive term in the form of a cubic nonlinearity is considered. On the basis of already known results, a more general formulation of the problem is investigated, with weaker sufficient conditions for the existence of a solution with an internal transition layer to be provided than in previous studies. For convenience, the known results are given, which ensure the fulfillment of the existence theorem of the contrast structure. The justification for the existence of a solution with an internal transition layer is based on the use of an asymptotic method of differential inequalities based on the modification of the terms of the constructed asymptotic expansion. Further, sufficient conditions are established to fulfill these requirements, and they have simple and concise formulations in the form of the algebraic equation $w(x_0,t) = 0$ and the condition $w_x(x_0,t) < 0$, which is essentially a condition of simplicity of the root $x_0(t)$ and ensuring the stability of the solution found. The function w is a function of the known functions appearing in the reactive and advective terms of the original problem. The equation $w(x_0,t) = 0$ is a problem for finding the zero approximation $x_0(t)$ to determine the localization region of the inner transition layer. In addition, the asymptotic Lyapunov stability of the found periodic solution is investigated, based on the application of the so-called compressible barrier method. The main result of the paper is formulated as a theorem.