  In this paper we study the Dirichlet problem for the
Kobayashi--Warren--Carter system. This system of parabolic PDE's models the
grain boundary motion in a polycrystal with a prescribed orientation at the
boundary of the domain. We obtain global existence in time of
energy-dissipative solutions. The regularity of the solutions as well as the
energy-dissipation property permit us to derive the steady-state problem as the
asymptotic in time limit of the system. We finally study the $\omega$-limit set
of the solutions; we completely characterize it in the one dimensional case,
showing, in particular that orientations in the $\omega$-limit set belong to
thee space of SBV functions. In the two dimensional case, we give sufficient
conditions for existence of radial symmetric piecewise constants solutions.
