  In this paper, global well-posedness of the non-Markovian Unruh-Zurek and
Hu-Paz-Zhang master equations with nonlinear electrostatic coupling is
demonstrated. They both consist of a Wigner-Poisson like equation subjected to
a dissipative Fokker-Planck mechanism with time-dependent coefficients of
integral type, which makes necessary to take into account the full history of
the open quantum system under consideration to describe its present state. From
a mathematical viewpoint this feature makes particularly elaborated the
calculation of the propagators that take part of the corresponding mild
formulations, as well as produces rather strong decays near the initial time
($t=0$) of the magnitudes involved, which would be reflected in the subsequent
derivation of a priori estimates and a significant lack of Sobolev regularity
when compared with their Markovian counterparts. The existence of local-in-time
solutions is deduced from a Banach fixed point argument, while global
solvability follows from appropriate kinetic energy estimates.
