  The final goal of this paper is to prove existence of local (strong)
solutions to a (fully nonlinear) porous medium equation with blow-up term and
nondecreasing constraint. To this end, the equation, arising in the context of
Damage Mechanics, is reformulated as a mixed form of two different types of
doubly nonlinear evolution equations. Global (in time) solutions to some
approximate problems are constructed by performing a time discretization
argument and by taking advantage of energy techniques based on specific
structures of the equation. Moreover, a variational comparison principle for
(possibly non-unique) approximate solutions is established and it also enables
us to obtain a local solution as a limit of approximate ones.
