Existence of solution for a generalized stochastic Cahn-Hilliard equationon convex domains

Dimitra Antonopoulou; Georgia Karali

doi:10.3934/dcdsb.2011.16.31

Article Contents

2011, Volume 16, Issue 1: 31-55. Doi: 10.3934/dcdsb.2011.16.31

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Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains

Dimitra Antonopoulou^1, and
Georgia Karali^2,

1.
Department of Applied Mathematics, University of Crete, Heraklion
2.
Department of Applied Mathematics, University Crete, P.O. Box 2208, 71409, Heraklion, Crete

Received: March 31, 2010

Revised: January 31, 2011

Published: June 2011

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Abstract

We consider a generalized Stochastic Cahn-Hilliard equation with multiplicative white noise posed on bounded convex domains in $R^d$, $d=1,2,3$, with piece-wise smooth boundary, and introduce an additive time dependent white noise term in the chemical potential. Since the Green's function of the problem is induced by a convolution semigroup, we present the equation in a weak stochastic integral formulation and prove existence of solution when $d\leq 2$ for general domains, and for $d=3$ for domains with minimum eigenfunction growth, without making use of any explicit expression of the spectrum and the eigenfunctions. The analysis is based on stochastic integral calculus, Galerkin approximations and the asymptotic spectral properties of the Neumann Laplacian operator. Existence is also derived for some non-convex cases when the boundary is smooth.

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Mathematics Subject Classification: 35K55, 35K40, 60H30, 60H15.

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