Two-parameter asymptotics in the Cauchy problem for a parabolic equation

TL;DR

This paper develops formal asymptotic solutions for a quasi-linear parabolic equation with two small parameters, addressing the complex interplay of initial conditions and higher derivatives in the Cauchy problem.

Contribution

It introduces a novel two-parameter asymptotic analysis approach for the Cauchy problem in quasi-linear parabolic equations with small parameters.

Findings

Constructed formal asymptotic solutions in small parameters

Analyzed the influence of initial step-like functions with small parameters

Provided insights into the behavior of solutions near initial discontinuities

Abstract

The Cauchy problem for a quasi-linear parabolic equation with a small parameter at a higher derivative is considered. The initial step-like function contains another small parameter. Formal asymptotic solutions of the problem in small parameters are constructed.