From weak discontinuity to gradient catastrophe

The Cauchy problem for a quasilinear parabolic equation with small parameter at the highest derivative is considered in the case when the solution of the degenerate equation has a weak discontinuity subsequently turning into a strong discontinuity. The singularities that the coefficients of the asymptotic formula representing the solution in the boundary layer of the weak discontinuity develop on approaching the point of the gradient catastrophe are analysed.