Stabilization of the front in a medium with discontinuous characteristics

We study the autowave front propagation in a medium with discontinuous characteristics and the conditions for its stabilization to a stationary solution with a large gradient at the interface between media in the one-dimensional case. The asymptotic method of differential inequalities, based on constructing an asymptotic approximation of the solution, is the main method of study. We develop an algorithm for constructing such an approximation for the solution of the moving front form in a medium with discontinuous characteristics. The application of such an algorithm requires a detailed analysis of the behavior of the solution in neighborhoods of two singular points: the front localization point and the medium discontinuity point. As a result, we obtain a system of equations for the front propagation speed; this distinguishes this paper from the previously published ones. The developed algorithm can be used to describe autowave propagation in layered media. The results can also be extended to the multidimensional case.