Stabilization of a moving front in a reaction-advection-diffusion problem

We consider an initial-boundary value problem for a one-dimensional reaction-diffusion-advection equation with a front-shaped solution. We study the stabilization of this front-shaped solution to a stationary solution of the corresponding problem. The proof of the stabilization theorem is based on the concepts of upper and lower solutions and consequences of comparison theorems. Upper and lower solutions with large gradients are constructed as modifications of the formal asymptotics of the moving front with respect to a small parameter. The main idea of ​​the proof is to show that the upper and lower solutions of the initial-boundary value problem fall within the basin of attraction of an asymptotically stable stationary solution over a sufficiently large time interval. The study conducted in this paper provides an answer to the question of the nonlocal basin of attraction of the stationary solution and allows us to derive some stationarity criteria. The results are illustrated with computational examples.