  Our studies are directed to the existence of weak solutions to a parabolic
problem containing a multi-valued term. The problem is formulated in the
language of maximal monotone graphs. We assume that the growth and coercivity
conditions of a nonlinear term are prescribed by means of time and space
dependent $N$--function. This results in formulation of the problem in
generalized Musielak-Orlicz spaces. We are using density arguments, hence an
important step of the proof is a uniform boundedness of appropriate convolution
operators in Musielak-Orlicz spaces. For this purpose we shall need to assume a
kind of logarithmic H\"older regularity with respect to $t$ and $x$.
