International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 178057, 27 pages
http://dx.doi.org/10.1155/2011/178057
Research Article

On the Dynamics of Nonautonomous Parabolic Systems Involving the Grushin Operators

Anh Cung The1 and Toi Vu Manh2

1Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, 10307 Hanoi, Vietnam
2Faculty of Computer Science and Engineering, Hanoi Water Resources University, 175 Tay Son, Dong Da, 10508 Hanoi, Vietnam

Received 19 December 2010; Accepted 21 February 2011

Academic Editor: Feng Qi

Copyright © 2011 Anh Cung The and Toi Vu Manh. 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 study the long-time behavior of solutions to nonautonomous semilinear parabolic systems involving the Grushin operators in bounded domains. We prove the existence of a pullback 𝒟 -attractor in ( 𝐿 2 ( Ω ) ) 𝑚 for the corresponding process in the general case. When the system has a special gradient structure, we prove that the obtained pullback 𝒟 -attractor is more regular and has a finite fractal dimension. The obtained results, in particular, extend and improve some existing ones for the reaction-diffusion equations and the Grushin equations.