  We consider an abstract parabolic problem in a framework of maximal monotone
graphs, possibly multi-valued with growth conditions formulated with help of an
$x-$dependent $N-$function. The main novelty of the paper consists in the lack
of any growth restrictions on the ${N}$--function combined with its anisotropic
character, namely we allow the dependence on all the directions of the
gradient, not only on its absolute value. This leads us to use the notion of
modular convergence and studying in detail the question of density of compactly
supported smooth functions with respect to the modular convergence.
