The Asymptotic Stability of a Stationary Solution with an Internal Transition Layer to a Reaction–Diffusion Problem with a Discontinuous Reactive Term

Abstract

The problem of the asymptotic stability of a stationary solution with an internal transition layer of a one-dimensional reaction–diffusion equation is considered. What makes this problem peculiar is that it has a discontinuity (of the first kind) of the reactive term (source) at an internal point of the segment on which the problem is stated, making the solutions have large gradients in the narrow transition layer near the interface. The existence, local uniqueness, and asymptotic stability conditions are obtained for the solution with such an internal transition layer. The proof uses the asymptotic method of differential inequalities. The obtained existence and stability conditions of the solution should be taken into account when constructing adequate models that describe phenomena in media with discontinuous characteristics. One can use the results of this work to develop efficient methods for solving differential equations with discontinuous coefficients numerically.