International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 875913, 30 pages
http://dx.doi.org/10.1155/2012/875913
Research Article

Pullback Attractors for Nonclassical Diffusion Equations in Noncylindrical Domains

Cung The Anh1 and Nguyen Duong Toan2

1Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
2Department of Mathematics, Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam

Received 26 March 2012; Accepted 20 May 2012

Academic Editor: Ram U. Verma

Copyright © 2012 Cung The Anh and Nguyen Duong Toan. 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

The existence and uniqueness of a variational solution are proved for the following nonautonomous nonclassical diffusion equation 𝑢 𝑡 − 𝜀 Δ 𝑢 𝑡 − Δ 𝑢 + 𝑓 ( 𝑢 ) = 𝑔 ( 𝑥 , 𝑡 ) , 𝜀 ∈ ( 0 , 1 ] , in a noncylindrical domain with homogeneous Dirichlet boundary conditions, under the assumption that the spatial domains are bounded and increase with time. Moreover, the nonautonomous dynamical system generated by this class of solutions is shown to have a pullback attractor 𝒜 𝜀 , which is upper semicontinuous at 𝜀 = 0 .