Let M be a complete Riemannian manifold which is either compact or has a pole, and let φ be a positive smooth function on M. In the warped product M ×φ r{double-struck}, we study the flow by the mean curvature of a locally Lipschitz continuous graph on M and prove that the flow exists for all time and that the evolving hypersurface is C ~∞ for t > 0 and is a graph for all t. Moreover, under certain conditions, the flow has a well-defined limit.
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