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Existence, uniqueness and stability of kinetic and entropy solutions to theboundary value problem for the Kolmogorov-type genuinely nonlinearultra-parabolic equation with a smooth source term is established. After this,we consider the case when the source term contains a small positive parameterand collapses to the Dirac delta-function, as this parameter tends to zero. Inthis case, the limiting passage from the original equation with the smoothsource to the impulsive ultra-parabolic equation is fulfilled and rigorouslyjustified. The proofs rely on the method of kinetic equation and on thecompensated compactness techniques for genuinely nonlinear equations.

Singular limits of the quasi-linear Kolmogorov-type equation with a source term