Pullback attractors for non-autonomous reaction–diffusion equation with infinite delays in Cγ,Lr(Ω) $C_{\gamma,L^{r}(\Omega)}$ or Cγ,W1,r(Ω) $C_{\gamma,W^{1,r}(\Omega)}

Abstract In this paper, the well-posedness for the non-autonomous reaction–diffusion equation with infinite delays on a bounded domain is established. The existence of pullback attractors for the process in Cγ,Lr(Ω) $C_{\gamma,L^{r}(\Omega)}$ and Cγ,W1,r(Ω) $C_{\gamma,W^{1,r}(\Omega)}$ is proved, respectively. The noncompact Kuratowski measure is applied to check the asymptotic compactness.