On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium

Abstract

The main result of the paper is the uniqueness of nonnegative solutions of the Cauchy problem and of the first and mixed boundary value problems for a class of degenerate parabolic equations which includes the model equation ut=(um)xx+(un)x, where m≥1 and n>0. In particular, n is allowed to be smaller than one. The proof is based on a refined test function argument. The condition that u be nonnegative is crucial, but the restriction to one space variable is not.