In this paper, we consider the nonclassical diffusion equation with memory and the nonlinearity of the polynomial growth condition of arbitrary order in the time-dependent space. First, the well-posedness of the solution for the equation is obtained in the time-dependent space $\mathscr{U}_{t}$
U
t
. Then, we establish the existence and regularity of the time-dependent global attractor. Finally, we also conclude that the fractal dimension of the time-dependent attractor is finite.