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### Open Access Journals

*Discrete and Continuous Dynamical Systems-A*is dedicated to Peter W. Bates on the occasion of his 60th birthday, and in recognition of his outstanding contributions to infinite dimensional dynamical systems and the mathematical theory of phase transitions.

Peter Bates was born in Manchester, England on December 27, 1947. He graduated from the University of London in mathematics in 1969 after which he moved to United States with his family. Later, he attended the University of Utah and received his Ph.D. in 1976. Following his graduation, Peter moved to Texas and taught at University of Texas at Pan American and Texas A&M University. He returned to Utah in 1984 and taught at Brigham Young University until 2004. He is currently a professor of mathematics at Michigan State University.

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In this paper, we study the limiting behavior of dynamics for stochastic reaction-diffusion equations driven by an additive noise and a deterministic non-autonomous forcing on an (*n*+1)-dimensional thin region when it collapses into an *n*-dimensional region. We first established the existence of attractors and their properties for these equations on (*n*+1)-dimensional thin domains. We then show that these attractors converge to the random attractor of the limit equation under the usual semi-distance as the thinness goes to zero.

In this paper, we establish the existence of random attractors for stochastic parabolic equations driven by additive noise as well as deterministic non-autonomous forcing terms in weighted Lebesgue spaces $ L_{\delta}^r(\mathcal{O})$, where $ 1<r<\infty ,\ \delta$ is the distance from $ x$ to the boundary. The nonlinearity $ f(x,u)$ of equation depending on the spatial variable does not have the bound on the derivative in $ u$, and then causes critical exponent. In both subcritical and critical cases, we get the well-posedness and dissipativeness of the problem under consideration and, by smoothing property of heat semigroup in weighted space, the asymptotical compactness of random dynamical system corresponding to the original system.

*local*solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which in particular includes white noise.

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