Existence of solution for a generalized stochastic Cahn-Hilliard equationon convex domains
Dimitra C. Antonopoulou, Georgia Karali · Discrete and Continuous Dynamical Systems - B · 2011
We consider a generalized Stochastic Cahn-Hilliard equation withmultiplicative white noise posed on bounded convex domains in$R^d$, $d=1,2,3$, with piece-wise smooth boundary, andintroduce an additive time dependent white noise term in thechemical potential. Since the Green's function of the problem isinduced by a convolution semigroup, we present the equation in aweak stochastic integral formulation and prove existence ofsolution when $d\leq 2$ for general domains, and for $d=3$ fordomains with minimum eigenfunction growth, without making use ofany explicit expression of the spectrum and the eigenfunctions.The analysis is based on stochastic integral calculus, Galerkinapproximations and the asymptotic spectral properties of theNeumann Laplacian operator. Existence is also derived for somenon-convex cases when the boundary is smooth.