Existence and Stability of the Solutions with Internal Layers in Multidimensional Problems of the Reaction-Diffusion-Advection Type with Balanced Nonlinearity

Abstract

In the present paper, we consider a multidimensional singularly perturbed problem for an elliptic equation referred to as the stationary reaction-diffusion-advection equation in applications. We formulate basic conditions of the existence of solutions with internal transition layers (contrust structures), and we construct an asymptotic approximation of an arbitrary-order accuracy to such solutions. We use a more efficient method for localizing the transition surface, which permits one to develop our approach to a more complicated case of balanced advection and reaction (the so-called critical case). 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 solutions, as stationary solutions of the corresponding parabolic problems.