  We address the problem of general dissipative regularization of the
quasilinear transport equation. We argue that the local behavior of solutions
to the regularized equation near the point of gradient catastrophe for the
transport equation is described by the logarithmic derivative of the Pearcey
function, a statement generalizing the result of A.M.Il'in. We provide some
analytic arguments supporting such conjecture and test it numerically.
