Anisotropic parabolic problems with slowly or rapidly growing terms

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.