  Gravitational theories differing from General Relativity may explain the
accelerated expansion of the Universe without a cosmological constant. However,
to pass local gravitational tests, a "screening mechanism" is needed to
suppress, on small scales, the fifth force driving the cosmological
acceleration. We consider the simplest of these theories, i.e. a scalar-tensor
theory with first-order derivative self-interactions, and study isolated
(static and spherically symmetric) non-relativistic and relativistic stars. We
produce screened solutions and use them as initial data for non-linear
numerical evolutions in spherical symmetry. We find that these solutions are
stable under large initial perturbations, as long as they do not cause
gravitational collapse. When gravitational collapse is triggered, the
characteristic speeds of the scalar evolution equation diverge, even before
apparent black-hole or sound horizons form. This casts doubts on whether the
dynamical evolution of screened stars may be predicted in these effective field
theories.
