The Cauchy problem for a class of nonlinear degenerate parabolic-hyperbolic equations

In this article, we prove a general existence theorem for a class of nonlinear degenerate parabolic-hyperbolic equations. Since the regions of parabolicity and hyperbolicity are coupled in a way that depends on the solution itself, there is almost no hope of decoupling the regions and then taking into account the parabolic and the hyperbolic features separately. The existence of solutions can be obtained by finding the limit of solutions for the regularized equation of strictly parabolic type. We use the energy methods and vanishing viscosity methods to prove the local existence and uniqueness of solution.