On strongly Petrovskii's parabolic SPDEs in arbitrary dimension

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 R^d which are parabolic in the sense of Petrovskii, and do not necessarily de ne 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