Dynamical behavior of nonclassical diffusion equations with the driving delay in time-dependent spaces

In this paper, we mainly investigate the asymptotic behavior of solutions for the nonclassical diffusion problem with hereditary effects and time-dependent perturbed parameter. The main novelty is that the delay term may be driven by a function under very minimal assumptions, namely, measurability and the fact that the phase space is a time-dependent space of functions that are continuous in time. The existence and regularity of time-dependent global attractors are proved by using a new analytical technique. It is remarkable that the nonlinearity $ f $ has no restriction on the upper growth.