Interacting dark energy with time varying equation of state and the
  $H_0$ tension
  Almost in all interacting dark energy models present in the literature, the
stability of the model becomes potentially sensitive to the dark energy
equation of state parameter $w_x$, and a singularity arises at `$w_x = -1$'.
Thus, it becomes mandatory to test the stability of the model into two separate
regions, namely, for quintessence and phantom. This essentially brings in a
discontinuity into the parameters space for $w_x$. Such discontinuity can be
removed with some specific choices of the interaction or coupling function. In
the present work we choose one particular coupling between dark matter and dark
energy which can successfully remove such instability and we allow a dynamical
dark energy equation of state parameter instead of the constant one. In
particular, considering a dynamical dark energy equation of state with only one
free parameter $w_0$, representing the current value of the dark energy
equation of state, we confront the interacting scenario with several
observational datasets. The results show that the present cosmological data
allow an interaction in the dark sector, in agreement with some latest claims
by several authors, and additionally, a phantom behaviour in the dark energy
equation of state is suggested at present. Moreover, for this case the tension
on $H_0$ is clearly released. As a final remark, we mention that according to
the Bayesian analysis, $\Lambda$-cold dark matter ($\Lambda$CDM) is always
favored over this interacting dark energy model.
