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The Cahn-Hilliard/Allen-Cahn equation with noise is a simplified mean fieldmodel of stochastic microscopic dynamics associated with adsorption anddesorption-spin flip mechanisms in the context of surface processes. For suchan equation we consider a multiplicative space-time white noise with diffusioncoefficient of sub-linear growth. Using technics from semigroup theory, weprove existence, and path regularity of stochastic solution depending on thatof the initial condition. Our results are also valid for the stochasticCahn-Hilliard equation with unbounded noise diffusion, for which previousresults were established only in the framework of a bounded diffusioncoefficient. We prove that the path regularity of stochastic solution dependson that of the initial condition, and are identical to those proved for thestochastic Cahn-Hilliard equation and a bounded noise diffusion coefficient. Ifthe initial condition vanishes, they are strictly less than 2-d/2 in space and1/2-d/8 in time. As expected from the theory of parabolic operators in thesense of Petrovski, the bi-Laplacian operator seems to be dominant in thecombined model.

Existence and regularity of solution for a Stochastic Cahn-Hilliard/Allen-Cahn equation with unbounded noise diffusion