  We advance a variational method to prove qualitative properties such as
symmetries, monotonicity, upper and lower bounds, sign properties, and
comparison principles for a large class of doubly-nonlinear evolutionary
problems including gradient flows, some nonlocal problems, and systems of
nonlinear parabolic equations.
  Our method is based on the so-called Weighted-Energy-Dissipation (WED)
variational approach. This consists in defining a global parameter-dependent
functional over entire trajectories and proving that its minimizers converge to
solutions to the target problem as the parameter goes to zero. Qualitative
properties and comparison principles can be easily proved for minimizers of the
WED functional and, by passing to the limit, for the limiting problem.
  Several applications of the abstract results to systems of nonlinear PDEs and
to fractional/nonlocal problems are presented. Eventually, we present some
extensions of this approach in order to deal with rate-independent systems and
hyperbolic problems.
