Abstract and Applied Analysis
Volume 2011 (2011), Article ID 616451, 21 pages
http://dx.doi.org/10.1155/2011/616451
Research Article

Existence of Random Attractors for a 𝑝 -Laplacian-Type Equation with Additive Noise

Wenqiang Zhao1,2 and Yangrong Li1

1School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received 10 January 2011; Revised 18 April 2011; Accepted 9 May 2011

Academic Editor: Yuming Shi

Copyright © 2011 Wenqiang Zhao and Yangrong Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We first establish the existence and uniqueness of a solution for a stochastic 𝑝 -Laplacian-type equation with additive white noise and show that the unique solution generates a stochastic dynamical system. By using the Dirichlet forms of Laplacian and an approximation procedure, the nonlinear obstacle, arising from the additive noise is overcome when we make energy estimate. Then, we obtain a random attractor for this stochastic dynamical system. Finally, under a restrictive assumption on the monotonicity coefficient, we find that the random attractor consists of a single point, and therefore the system possesses a unique stationary solution.