  We study the gradient flow of the potential energy on the
infinite-dimensional Riemannian manifold of spatial curves parametrized by the
arc length, which models overdamped motion of a falling inextensible string. We
prove existence of generalized solutions to the corresponding nonlinear
evolutionary PDE and their exponential decay to the equilibrium. We also
observe that the system admits solutions backwards in time, which leads to
non-uniqueness of trajectories.
