On the asymptotics of the solution to the Cauchy problem for a singularly perturbed system of transfer equations with low nonlinear diffusion

This paper is a survey of results concerning asymptotics of solutions of singularly perturbed systems of transport equations; it also contains some new results. We discuss the so-called critical problems whose degenerate solutions are one-parameter families. Under certain conditions, this leads to a fast establishment of dynamic equilibrium between the components of the solution and the subsequent transfer with an “average” rate. The regions of large gradients of the initial conditions generate inner layers, which can be described by linear parabolic equations and their generalizations, for example, equations of the Burgers and Burgers–Korteweg–de Vries types.