Strong attractors for the nonclassical diffusion equation with fading memory in time-dependent spaces

In this paper, we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term $f$ fulfills the polynomial growth of arbitrary order and the external force $g(x)\in L^{2}(\Omega)$. In the framework of time-dependent spaces, we verify the existence and uniqueness of strong solutions by the Galerkin method, then we obtain the existence of the time-dependent global attractor $\mathscr{A}=\{A_t\}_{t\in \mathbb{R}}$ in $\mathcal{M}_t^1$.