Convergence result and blow-up examples for the Guan-Li mean curvature flow on warped product spaces

We examine the question of convergence of solutions to a geometric flow which was introduced by Guan and Li [6] for starshaped hypersurfaces in space forms and generalized by Guan, Li, and Wang [7] to the case of warped product spaces. We obtain a convergence result under a condition on the optimal modulus of continuity of the initial data. Moreover we show by examples that this condition is optimal at least in the one-dimensional case.