  Here, we study a level-set forced mean curvature flow with the homogeneous
Neumann boundary condition. We first show that the solution is Lipschitz in
time and locally Lipschitz in space. Then, under an additional condition on the
forcing term, we prove that the solution is globally Lipschitz. We obtain the
large time behavior of the solution in this setting and study the large time
profile in some specific situations. Finally, we give two examples
demonstrating that the additional condition on the forcing term is sharp, and
without it, the solution might not be globally Lipschitz.
