  We consider a stochastic partial differential equation with logarithmic (or
negative power) nonlinearity, with one reflection at 0 and with a constraint of
conservation of the space average. The equation, driven by the derivative in
space of a space-time white noise, contains a bi-Laplacian in the drift. The
lack of the maximum principle for the bi-Laplacian generates difficulties for
the classical penalization method, which uses a crucial monotonicity property.
Being inspired by the works of Debussche and Zambotti, we use a method based on
infinite dimensional equations, approximation by regular equations and
convergence of the approximated semi-group. We obtain existence and uniqueness
of solution for nonnegative intial conditions, results on the invariant
measures, and on the reflection measures.
