Level-set forced mean curvature flow with the Neumann boundary condition

We investigate a level-set forced mean curvature flow with homogeneous Neumann boundary conditions. The well-posedness and the comparison principle of this flow have been well-established in the theory of viscosity solutions. However, our objective is to go beyond the well-posedness theory and gain insight into the long-term behavior of the solution. We begin by demonstrating that the solution is Lipschitz in time and locally Lipschitz in space. Furthermore, by introducing an additional condition on the forcing term, we prove that the solution is globally Lipschitz. Leveraging this result, we obtain the large-time behavior of the solution in this context.