On The Existence and Asymptotic Stability of Periodic Contrast Structures in Quasilinear Reaction-Advection-Diffusion Equations

Abstract

We consider periodic solutions with internal transition and boundary layers (periodic contrast structures) for a singularly perturbed parabolic equation that is referred to in applications as reaction-advection-diffusion equation. An asymptotic approximation to such solutions is constructed and an existence theorem is proved. An efficient algorithm is developed for constructing an asymptotic approximation to the localization curve of the transition layer. To substantiate the asymptotic thus constructed, we use the asymptotic method of differential inequalities. Moreover, we assert that asymptotic stability of the solution in the sense of Lyapunov occurs.