A kinetic approach to comparison properties for degenerate parabolic–hyperbolic equations with boundary conditions

We study the comparison principle for degenerate parabolic–hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and supersolution. The L1 contractivity and, therefore, uniqueness of entropy solutions has been obtained so far by some authors, but it seems that any comparison theorem is not proven. The method used there is the doubling variable technique due to Kružkov. Our method is based upon the kinetic formulation and the kinetic techniques. By developing the kinetic techniques for degenerate parabolic–hyperbolic equations with boundary conditions, we can obtain a comparison property which obviously extends the L1 contractive property.