  A possible candidate for the late time accelerated expanding Universe is
phantom energy, which possesses rather bizarre properties, such as the
prediction of a Big Rip singularity and the violation of the null energy
condition. The latter is a fundamental ingredient of traversable wormholes, and
it has been shown that phantom energy may indeed sustain these exotic
geometries. Inspired by the evolving dark energy parameter crossing the phantom
divide, we consider in this work a varying equation of state parameter
dependent on the radial coordinate, i.e., $\omega(r)=p(r)/\rho(r)$. We shall
impose that phantom energy is concentrated in the neighborhood of the throat,
to ensure the flaring out condition, and several models are analyzed. We shall
also consider the possibility that these phantom wormholes be sustained by
their own quantum fluctuations. The energy density of the graviton one loop
contribution to a classical energy in a phantom wormhole background and the
finite one loop energy density are considered as a self-consistent source for
these wormhole geometries. The latter semi-classical approach prohibits
solutions with a constant equation of state parameter, which further motivates
the imposition of a radial dependent parameter, $\omega(r)$, and only permits
solutions with a steep positive slope proportional to the radial derivative of
the equation of state parameter, evaluated at the throat. The size of the
wormhole throat as a function of the relevant parameters is also explored.
