Large-Time Behavior for a Phase-Field System of 3D-Grain Boundary Motion

In this paper, we study a three dimensional model for grain boundary motion with the presence of time-dependent external forces. We show existence of solutions in the large. We also show that the 𝜔-limit set of the solutions is compact and that the 𝜔-limit points satisfy the corresponding elliptic system. In the case of no external forcing for the rotation and, under a condition on the range of the initial datum, we prove that the system reaches a steady state in a finite time and that, in this case, the rotation becomes a constant one.