Regularity of pullback attractors for nonclassical diffusion equations with delay

In this paper, we mainly study the regularity of pullback $\mathcal{D}$-attractors for a nonautonomous nonclassical diffusion equation with delay term $b(t,u_t)$ which contains some hereditary characteristics. Under a critical nonlinearity $f$, a time-dependent force $g(t,x)$ with exponential growth and a delayed force term $b(t,u_t)$, we prove that there exists a pullback $\mathcal{D}$-attractor $\mathcal{A}=\{A(t):t\in \mathbb{R}\}$ in ${\mathbb{K}}^1=H_0^1(\mathrm{\Omega })\times L^2((-h,0);L^2(\mathrm{\Omega }))$ to problem \eqref{ine01} and for each $t\in \mathbb{R}$, $A(t)$ is bounded in ${\mathbb{K}}^2=H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\times L^2((-h,0);L^2(\mathrm{\Omega }))$.