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

Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem associated with the Kolmogorov-type, genuinely nonlinear, degenerate hyperbolic-parabolic (ultra-parabolic) equation with a smooth source term is established. In addition, we consider the case when the source term contains a small positive parameter and collapses to the Dirac delta-function, as this parameter tends to zero. In this case, the limiting passage from the original equation with the smooth source to the impulsive ultra-parabolic equation is investigated and the formal limit is rigorously justified. Our proofs rely on the use of kinetic equations and the compensated compactness method for genuinely nonlinear balance laws.