We consider a stochastic partial differential equation with two logarithmic
nonlinearities, with two reflections at 1 and -1 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, Gouden\`ege and Zambotti, we obtain
existence and uniqueness of solution for initial conditions in the interval
(−1,1). Finally, we prove that the unique invariant measure is ergodic, and
we give a result of exponential mixing
