  We propose an extension of the classical variational theory of evolution
equations that accounts for dynamics also in possibly non-reflexive and
non-separable spaces. The pivoting point is to establish a novel variational
structure, based on abstract modular spaces associated to a given convex
function. Firstly, we show that the new variational triple is suited for
framing the evolution, in the sense that a novel duality paring can be
introduced and a generalised computational chain rule holds. Secondly, we prove
well-posedness in an extended variational sense for evolution equations,
without relying on any reflexivity assumption and any polynomial requirement on
the nonlinearity. Finally, we discuss several important applications that can
be addressed in this framework: these cover, but are not limited to, equations
in Musielak-Orlicz-Sobolev spaces, such as variable exponent, Orlicz, weighted
Lebesgue, and double-phase spaces.
