Existence and Stability of Contrast Structures in Multidimensional Singularly Perturbed Reaction-Diffusion-Advection Problems

Abstract

We consider stationary solutions with boundary and internal transition layers (contrast structures) for a nonlinear singularly perturbed equation that is referred to in applications as the stationary reaction-diffusion-advection equation. We construct an asymptotic approximation of an arbitrary-order accuracy to such solutions and prove the existence theorem. We suggest an afficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed asymptotics, we use and develop, to this class of problems, an asymptotic method of differential inequalities, which also permits one to prove the Lyapunov stability of such stationary solutions. The results can be used to create the numerical method which uses the asymptotic analyses to create space non uniform meshes to describe internal layer behavior of the solution.