  This paper is concerned with a parabolic evolution equation of the form
$A(u_t) + B(u) = f$, settled in a smooth bounded domain of ${\bf R}^d$, $d \geq
1$, and complemented with the initial conditions and with (for simplicity)
homogeneous Dirichlet boundary conditions. Here, $-B$ stands for a diffusion
operator, possibly nonlinear, which may range in a very wide class, including
the Laplacian, the $m$-Laplacian for suitable $m\in(1,\infty)$), the
"variable-exponent" $m(x)$-Laplacian, or even some fractional order operators.
The operator $A$ is assumed to be in the form $[A(v)](x, t) = \alpha(x, v(x,
t))$ with $\alpha$ being measurable in $x$ and maximal monotone in $v$. The
main results are devoted to proving existence of weak solutions for a wide
class of functions $\alpha$ that extends the setting considered in previous
results related to the variable exponent case where $\alpha(x, v) =
|v(x)|^{p(x)-2} v(x)$. To this end, a theory of subdifferential operators will
be established in Musielak-Orlicz spaces satisfying structure conditions of the
so-called $\Delta_2$-type and a framework for approximating maximal monotone
operators acting in that class of spaces will also be developed. Such a theory
is then applied to provide an existence result for a specific equation, but it
may have an independent interest in itself. Finally, the existence result is
illustrated by presenting a number of specific equations (and, correspondingly,
of operators $A$, $B$) to which the result can be applied.
