On critical behavior in nonlinear evolutionary PDEs with small viscosity

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.