  In this paper we show that the Cahn-Hilliard stochastic SPDE has a function
valued solution in dimension 4 and 5 when the perturbation is driven by a
space-correlated Gaussian noise. This is done proving general results on SPDEs
with globally Lipschitz coefficients associated with operators on smooth
domains of $\mathbb{R}^d$ which are parabolic in the sense of Petrovskii}, and
do not necessarily define a semi-group of operators. We study the regularity of
the trajectories of the solutions and the absolute continuity of the law at
some given time and position.
