  This paper is concerned with the Cauchy-Dirichlet problem for a doubly
nonlinear parabolic equation involving variable exponents and provides some
theorems on existence and regularity of strong solutions. In the proof of these
results, we also analyze the relations occurring between Lebesgue spaces of
space-time variables and Lebesgue-Bochner spaces of vector-valued functions,
with a special emphasis on measurability issues and particularly referring to
the case of space-dependent variable exponents. Moreover, we establish a chain
rule for (possibly nonsmooth) convex functionals defined on variable exponent
spaces. Actually, in such a peculiar functional setting the proof of this
integration formula is nontrivial and requires a proper reformulation of some
basic concepts of convex analysis, like those of resolvent, of Yosida
approximation, and of Moreau-Yosida regularization.
