LOCAL EXISTENCE AND UNIQUENESS OF SOLUTIONS OF DEGENERATE PARABOLIC EQUATIONS

We consider a class of degenerate parabolic equations on a bounded domain with mixed boundary conditions. These problems arise, for example, in the study of flow through porous media. Under appropriate hypotheses, we establish the existence of a nonnegative solution which is obtainable as a monotone limit of solutions of quasilinear parabolic equations. This construction is used to establish uniqueness, comparison, and L1 continuous dependence theorems, as well as some results on blow up of solutions in finite time.