1. Introduction. In recent years, Conti, Pata and Temam[10] introduced the concep-tion of time-dependent global attractors. Based on this new theoretical framework, we are able to study the long-term behavior of solutions for the non-autonomous evolution equations on time-dependent spaces. The non-autonomous nature of the systems is characterized by the spaces rather than the external forcing terms. For the long-term behavior of solutions on time-dependent spaces, especially for the weak solutions, there are some literatures, see, e.g., [10, 7, 6, 17, 18, 22, 19, 16, 24, 27, 23, 26, 31, 52, 37...
