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

We define a new class of admissible solutions for the parabolic problems wh ere the theory of viscosity solutions does not apply. A typical example is the porous medium equation complemented by lower-order terms, nonlinear wit h respect to the gradient. In the case, the nonlinearity in the equation fa il to be proper in the sense of the theory of viscosity solutions. The admi ssible solution satisfy a very general comparison principle and, consequent ly, the corresponding initial-boundary value problems are well-posed in thi s class. They coincide with the standard viscosity solutions provided the n onlinearity in the equation is proper.