Abstract
This paper is devoted to the study of mild solutions for the initial and boundary value problem of stochastic viscous Cahn-Hilliard equation driven by white noise. Under reasonable assumptions we first prove the existence and uniqueness result. Then, we show that the existence of a stochastic global attractor which pullback attracts each bounded set in appropriate phase spaces.
Highlights
The model was first introduced by Novick-Cohen 1 to describe the dynamics of viscous first order phase transitions, which has been extensively studied in the past decades
The main results of the paper are contained in the following two theorems
While the deterministic model captures more intrinsic nature of phase transitions in binary, it ignores some random effects such as thermal fluctuations which are present in any material
Summary
This paper is devoted to the existence of mild solutions and global asymptotic behavior for the following stochastic viscous Cahn-Hilliard equation:. Instead of deterministic viscous Cahn-hilliard equation, here, we consider the general stochastic equation 1.1 which is affected by a space-time white noise In such a case, new difficulties appear, and the resulting stochastic model must be treated in a different way. Crauel and Flandoli 7 see Schmalfuss 8 introduced the concept of a random attractor as a proper generalization of the corresponding deterministic global attractor which turns out to be very helpful in the understanding of the long-time dynamics for stochastic differential equations In this present work, we first establish some existence results on mild solutions. In case α 0, 1.1 reduces to the stochastic Cahn-Hilliard equation which was studied in 9 , where the authors obtain the existence and uniqueness of the weak solutions to the initial and Neumann boundary value problem in some phase spaces under appropriate assumptions on noise. The last section stands as an appendix for some basic knowledge of random dynamical system RDS
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