Asymptotics of solutions of a parabolic equation near singular points

Abstract

Results of investigation of the asymptotic behavior of solutions to the Cauchy problems for a quasi-linear parabolic equation with a small parameter at a higher derivative near singular points of limit solutions are presented. Interest to the problem under consideration is explained by its applications to a wide class of physical systems and probabilistic processes such as acoustic waves in fluid and gas, hydrodynamical turbulence and nonlinear diffusion. The following cases are considered: a singularity generated by a jump discontinuity of the initial function, collision of two shock waves, gradient catastrophe, transition of a weak discontinuity into a shock wave, a singularity generated by a large initial gradient.