Attractors for the nonclassical diffusion equations with fading memory

We discuss long-time dynamical behavior of the nonclassical diffusion equation with fading memory when nonlinearity is critical. The existence and regularity of global attractors in weak topological space and strong topological space are obtained, while the forcing term only belongs to H−1(Ω) and L2(Ω) respectively. The results in this part are new and appear to be optimal corresponding to the forcing term.