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

Wenqiang Zhao (School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, China)

Yangrong Li (School of Mathematics and Statistics, Southwest China University, Chongqing, China)

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

Full Text (PDF format)