Cahn-Hilliard stochastic equation: existence of the solution and of its density

We show the existence and uniqueness of a function-valued process solution to the stochastic Cahn-Hilliard equation driven by space-time white noise with a nonlinear diffusion coefficient. Then we show that the solution is locally differentiable in the sense of the Malliavin calculus, and, under some non-degeneracy condition on the diffusion coefficient, that the law of the solution is absolutely continuous with respect to Lebesgue measure.