  Let $M$ be a complete Riemannian manifold which either is compact or has a
pole, and let $\varphi$ be a positive smooth function on $M$. In the warped
product $M\times_\varphi\mathbb R$, 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 $C^\infty$ for $t>0$ and is a
graph for all $t$. Moreover, under certain conditions, the flow has a well
defined limit.
