  In this paper we study a variational system of two parabolic PDEs, called the
Kobayashi-Warren-Carter system, which models the grain boundary motion in a
polycrystal. The focus of the study is the existence of solutions to this
system which dissipate the associated energy functional. We obtain existence of
this type of solutions via a suitable approximation of the energy functional
with Laplacians and an extra regularization of the weighted total variation
term of the energy. As a byproduct of this result, we also prove some
$\Gamma$-convergence results concerning weighted total variations and the
corresponding time-dependent cases. Finally, the regularity obtained for the
solutions together with the energy dissipation property, permits us to
completely characterize the $\omega$-limit set of the solutions.
