Asymptotic Solutions of a Parabolic Equation Near Singular Points of A and B Types

Abstract

The Cauchy problem for a quasi-linear parabolic equation with a small parameter multiplying a higher derivative is considered in two cases when the solution of the limit problem has a point of gradient catastrophe. Asymptotic solutions are found by using the Cole–Hopf transform. The integrals determining the asymptotic solutions correspond to the Lagrange singularities of type A and the boundary singularities of type B. The behavior of the asymptotic solutions is described in terms of the weighted Sobolev spaces.