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

Kobayashi--Warren--Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation