Remarks on the generalized Cauchy-Dirichlet problem for graph mean curvature flow with driving force

We study a generalized Cauchy-Dirichlet problem for graph forced mean curvature flow equations in the sense of viscosity solutions. It is well-known that viscosity solutions of Cauchy-Dirichlet problems may not satisfy the boundary condition pointwise in general. We prove that if viscosity solutions lose the Dirichlet boundary condition, then the solution satisfies the singular Neumann boundary condition. This fact causes a difficulty to study the large-time behavior. In this paper, we prove that the viscosity solution to a generalized Cauchy-Dirichlet problem converges to the appropriate traveling wave type solution when the domain is a N-dimensional ball.