  In this paper, we consider three phantom dark energy models, in the context
of interaction between the dark components namely cold dark matter (CDM) and
dark energy (DE). The first model, known as $w_{\textrm{d}}$CDM can induce a
big rip singularity (BR) while the two remaining induce future abrupt events
known as the Little Rip (LR) and Little Sibling of the Big Rip (LSBR). These
phantom DE models can be distinguished by their equation of state. We invoke a
new phenomenon such as the interaction between CDM and DE given that it could
solve or alleviate some of the problems encountered in standard cosmology. We
aim to find out the effect of such an interaction on the cosmological
parameters of the studied models, as well as, the persistence or the
disappearance of the singularity and the abrupt events induced by the models
under study. We choose an interaction term proportional to DE density, i.e.
$Q=\lambda H \rho_{\textrm{d}}$, since the case where $Q\propto
\rho_{\textrm{m}}$ could lead to a large scale instability at early time. We
also do not claim at all that $Q=\lambda H \rho_{\textrm{d}}$ is the ideal
choice since it suffers from a negative CDM density in the future. By the use
of a Markov Chain Monte Carlo (MCMC) approach, and by assuming a flat FLRW
Universe, we constrain the cosmological parameters of each of the three phantom
DE models studied. Furthermore, by the aid of the corrected Akaike Information
Criterion ($\text{AIC}_{c}$) tool, we compare our phantom DE models. Finally, a
perturbative analysis of phantom DE models under consideration is performed
based on the best fit background parameters.
