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In this paper, we study the forced mean curvature flows and the prescribedmean curvature equations of both graphs and level-sets with capillary-typeboundary conditions on a $C^3$ bounded domain, which is not necessarily convex.We prove a priori gradient estimates locally Lipschitz in time. Under anassumption on the forcing term, we prove that the gradient estimates areglobally Lipschitz in time. As a consequence, we obtain the existence theoremof solutions. In our formulation, we recover the known results of the gradientestimates on a strictly convex $C^3$ bounded domain. Next, we study theassociated eigenvalue problems for mean curvature flows of both graphs andlevel-sets. We prove the large time behavior of the solutions of mean curvatureflows of graphs on a smooth bounded domain. Finally, we compute the asymptoticspeed of the solutions of level-set mean curvature flows and the large timeprofile of level-sets in the radially symmetric case based on optimal controlformula. Examples arising in the radially symmetric case demonstrate that theadditional assumption on the forcing term is optimal.

Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain