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.

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In this paper, we study the forced mean curvature flows and the prescribed mean curvature equations of both graphs and level-sets with capillary-type boundary conditions on a C 3 bounded domain, which is not necessarily convex. We prove a priori gradient estimates locally Lipschitz in time. Under an assumption on the forcing term, we prove that the gradient estimates are globally Lipschitz in time. As a consequence, we obtain the existence theorem of solutions. In our formulation, we recover the known results of the gradient estimates on a strictly convex C 3 bounded domain. Next, we study the associated eigenvalue problems for mean curvature flows of both graphs and level-sets. We prove the large time behavior of the solutions of mean curvature flows of graphs on a smooth bounded domain. Finally, we compute the asymptotic speed of the solutions of level-set mean curvature flows and the large time profile of level-sets in the radially symmetric case based on optimal control formula. Examples arising in the radially symmetric case demonstrate that the additional assumption on the forcing term is optimal.