Pullback attractors for nonclassical diffusion equations with a delay operator

In this paper, we consider the asymptotic behavior of weak solutions for nonclassical non-autonomous diffusion equations with a delay operator in time-dependent spaces when the nonlinear function $g$ satisfies subcritical exponent growth conditions, the delay operator $\phi (t,u_t)$ contains some hereditary characteristics and the external force $k\in L_{loc}^2(\mathbb{R};L^2(\mathrm{\Omega }))$. First, we prove the well-posedness of solutions by using the Faedo-Galerkin approximation method. Then after a series of elaborate energy estimates and calculations, we establish the existence and regularity of pullback attractors in time-dependent spaces $C_{{\mathcal{H}}_t(\mathrm{\Omega })}$ and $C_{{\mathcal{H}}_t^1(\mathrm{\Omega })}$, respectively.