<p>We consider nonlinear parabolic problems in a variational framework. The leading part is a monotone operator whose growth is controlled by time- and space-dependent Musielak functions. On Musielak’s controlling functions we impose regularity conditions which make it possible to extend certain classical results such as the density of smooth functions, a Poincaré-type inequality, an integration-by-parts formula and a trace result. Bringing together these results, we adapt the classical theory of monotone operators and prove the well-posedness of the variational problem.</p>