Viscosity solutions for quasilinear degenerate parabolic equations of porous medium type

Abstract

We consider the Cauchy Problem for the class of nonlinear parabolic equations of the form u t = a(u)Δu + |⊇u| 2 , with a function a(u) that vanishes at u = 0. Because of the degenerate character of the coefficient a the usual concept of viscosity solution in the sense of Crandall-Evans-Lions has to be modified to include the behaviour at the free boundary. We prove that the problem is well-posed in a suitable class of viscosity solutions. Agreement with the concept of weak solution is also shown.