Pullback Attractors for Nonclassical Diffusion Equations With a Delay Operator

In this paper, we consider the asymptotic behavior of weak solutions for nonclassical nonautonomous 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 $k\in L_{loc}^2\left(R,;,L^2,\left(\Omega \right)\right)$. 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 $C_{H_t\left(\Omega \right)}$ and $C_{H_t^1\left(\Omega \right)}$, respectively.