Time-dependent matter instability and star singularity in $F(R)$ gravity
  We investigate a curvature singularity appearing in the star collapse process
in $F(R)$ gravity. In particular, we propose an understanding of the mechanism
to produce the curvature singularity. Moreover, we explicitly demonstrate that
$R^\alpha$ ($1 < \alpha \leq 2$) term addition could cure the curvature
singularity and viable $F(R)$ gravity models could become free of such a
singularity. Furthermore, we discuss the realization process of the curvature
singularity and estimate the time scale of its appearance. For exponential
gravity, it is shown that in case of the star collapse, the time scale is much
shorter than the age of the universe, whereas in cosmological circumstances, it
is as long as the cosmological time.
