  We study a class of reflected McKean-Vlasov diffusions over a convex domain
with self-stabilizing coefficients. This includes coefficients that do not
satisfy the classical Wasserstein Lipschitz condition. Further, the process is
constrained to a (not necessarily bounded) convex domain by a local time on the
boundary. These equations include the subclass of reflected self-stabilizing
diffusions that drift towards their mean via a convolution of the solution law
with a stabilizing potential.
  Firstly, we establish existence and uniqueness results for this class and
address the propagation of chaos. We work with a broad class of coefficients,
including drift terms that are locally Lipschitz in spatial and measure
variables. However, we do not rely on the boundedness of the domain or the
coefficients to account for these non-linearities and instead use the
self-stabilizing properties.
  We prove a Freidlin-Wentzell type Large Deviations Principle and an
Eyring-Kramer's law for the exit-time from subdomains contained in the interior
of the reflecting domain.
