Asymptotics of solving a singularly perturbed system of transport equations with fast and slow components in the critical case

TL;DR

This paper develops an asymptotic expansion for a system of three transfer equations with both fast and slow components in a critical degenerate case, addressing the complex behavior of solutions as a small parameter approaches zero.

Contribution

It introduces a novel asymptotic expansion method for a critical singularly perturbed system of transfer equations with mixed components.

Findings

Asymptotic expansion combines regular and boundary parts.

Addresses critical case where the degenerate problem has a one-parameter family.

Provides a framework for analyzing solutions with smooth initial conditions.

Abstract

An asymptotic expansion with respect to a small parameter of the solution of the Cauchy problem is constructed for a system of three transfer equations, two of which are singularly perturbed by the degeneracy of the entire senior part of the transfer operator, and the third equation clearly does not contain a small parameter. The peculiarity of the problem is that it belongs to the so-called critical case: the solution of a degenerate problem is a one-parameter family. The asymptotic expansion of the solution under smooth initial conditions is constructed as the sum of the regular part and boundary functions.