IUMJ

Title: Viscosity solutions for quasilinear degenerate parabolic equations of porous medium type

Authors: Juan Luis Vazquez and Cristina Brandle

Issue: Volume 54 (2005), Issue 3, 817-860

Abstract:

We consider the Cauchy Problem for the class of nonlinear parabolic equations of the form $$u_t=a(u)\Delta u+|\nabla u|^2,$$ 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.