Admissible solutions for a class of nonlinear parabolic problems with non-negative data

Abstract

We define a new class of admissible solutions for the parabolic problems where the theory of viscosity solutions does not apply. A typical example is the porous medium equation complemented by lower-order terms, nonlinear with respect to the gradient. In the case, the nonlinearity in the equation fails to be proper in the sense of the theory of viscosity solutions. The admissible solutions satisfy a very general comparison principle and, consequently, the corresponding initial-boundary value problems are well-posed in this class. They coincide with the standard viscosity solutions provided the nonlinearity in the equation is proper.