Interacting realization of cosmological singularities with variable
  vacuum energy
  We examine an interacting dark matter--variable vacuum energy model for a
spatially flat Friedmann-Roberston-Walker spacetime, focusing on the appearance
of cosmological singularities such as \emph{big rip, big brake, big freeze},
and \emph{ big separation} along with abrupt events (\emph{infinite $\gamma$-
singularity} and \emph{new w-singularity}) at late times. We introduce a
phenomenological interaction which has a nonlinear dependence on the total
energy density of the dark sector and its derivative, solve exactly the source
equation for the model and find the energy density as function of the scale
factor as well as the time dependence of the approximate scale factor in the
neighborhood of the singularities. We describe the main characteristics of
these singularities by exploring the type of interaction that makes them
possible along with behavior of dark components near them. We apply the
geometric Tipler and Kr\'olak method for determining the fate of time-like
geodesic curves around the singularities. We also explore the strength of them
by analyzing the leading term in some geometric invariants such as the square
Riemann scalar and the Ricci scalar.
