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.

Keywords

• Quasi-linear parabolic equation, Cole–Hopf transform, Singular points, Asymptotic solutions, Whitney fold singularity, Il'in's universal solution, Weighted Sobolev spaces