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# Dynamics of Partial Differential Equations

## Volume 11 (2014)

### Number 3

### Random attractors for stochastic semi-linear degenerate parabolic equations with additive noises

Pages: 269 – 298

DOI: http://dx.doi.org/10.4310/DPDE.2014.v11.n3.a4

#### Authors

#### Abstract

The existences of random attractors in $L^p(D_N) \cap L^{2p-2} (D_N)$ are proved for a class of stochastic semi-linear degenerate parabolic equations on arbitrary bounded or unbounded domains $D_N \subseteq \mathbb{R}^N$, where the leading term of the equations has the form $\mathrm{div}(\sigma(x)\nabla u)$ and the nonlinearity $f(x, u)$ satisfies some dissipative assumptions and the growth of order $p-1, p \gt 2$. The asymptotic compactness of the corresponding random dynamical system in $L^p(D_N)$ and $L^{2p-2} (D_N)$ are established respectively by using an asymptotic a priori estimate method. Our result improves a previous result of Yang and Kloeden [25] concerning the existence of a compact random attractor in $L^2(D_N)$ for the same equations.

#### Keywords

random dynamical systems, stochastic semi-linear degenerate parabolic equation, asymptotic compactness, random attractor

#### 2010 Mathematics Subject Classification

35B40, 35B41, 35R60