Asymptotic solutions of a parabolic equation near singular points of $A$ and $B$ types

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.