Attractors for the non-autonomous nonclassical diffusion equations with fading memory

The long-time dynamical behavior of the non-autonomous nonclassical diffusion equation with fading memory, when nonlinearity is critical, is discussed for in the weak topological space H01(Ω)×Lμ2(R+;H01(Ω)). First, the asymptotic regularity of solutions is proven, and then the existence of a compact uniform attractor together with its structure and regularity is obtained, while the time-dependent forcing term is only translation bounded instead of translation compact. The result extends and improves some results given in [Y. Xiao, Attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. Engl. Ser. 18 (2002) 273–276; C. Sun, M. Yang, Dynamics of the nonclassical diffusion equations, Asympt. Anal. 59 (2008) 51–81].