Electron. J. Diff. Equ., Vol. 2012 (2012), No. 207, pp. 1-22.

Regularity of random attractors for stochastic semilinear degenerate parabolic equations

Cung The Anh, Tang Quoc Bao, Nguyen Van Thanh

Abstract:
We consider the stochastic semilinear degenerate parabolic equation
$$ du+[- \operatorname{div}(\sigma(x)\nabla u) + f(u) + \lambda u]dt = gdt+ \sum_{j=1}^{m}h_j{d\omega_j} $$
in a bounded domain $\mathcal{O}\subset \mathbb {R}^N$, with the nonlinearity satisfies an arbitrary polynomial growth condition. The random dynamical system generated by the equation is shown to have a random attractor $\{\mathcal{A}(\omega)\}_{\omega\in\Omega}$ in $\mathcal D_0^1(\mathcal{O},\sigma)\cap L^p(\mathcal{O})$. The results obtained improve some recent ones for stochastic semilinear degenerate parabolic equations.

Submitted September 27, 2012. Published November 25, 2012.
Math Subject Classifications: 35B41, 37H10, 35K65.
Key Words: Random dynamical systems; random attractors; regularity; stochastic degenerate parabolic equations; asymptotic a priori estimate method.

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Cung The Anh
Department of Mathematics
Hanoi National Univiersity of Education
136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
email: anhctmath@hnue.edu.vn
Tang Quoc Bao
School of Applied Mathematics and Informatics
Hanoi University of Science and Technology
1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam
email: baotangquoc@gmail.com
Nguyen Van Thanh
Foreign Languages Specialized School
University Of Languages and International Studies
Hanoi National University
2 Pham Van Dong, Cau Giay, Hanoi, Vietnam
email: thanhnvmath@gmail.com

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