Nonlinear parabolic problems in Musielak–Orlicz spaces

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 a time and space dependent NN-function. This results in the 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 Hölder regularity with respect to tt and xx.