Mean Curvature Type Flows of Graphs in Product Manifolds

In this note we study a large class of mean curvature type flows of graphs in product manifold $N\times R$ where N is a closed Riemann- ian manifold. Their speeds are the mean curvature of graphs plus a prescribed function. We establish long time existence and uniformly convergence of those flows with a barrier condition and a condition on the derivative of prescribed function with respect to the height. As an application we construct a weighted mean curvature flow in large classes of warped product manifolds which evolves each graph into a totally ge- odesic slice