Reaction–diffusion equations with non-autonomous force in H−1H−1 and delays under measurability conditions on the driving delay term

In this paper we analyze the existence of solutions for a reaction–diffusion problem with hereditary effects and a time-dependent force term with values in H−1H−1. The main novelty is that the delay term may be driven by a function under very minimal assumptions, namely, just measurability. This is due to the fact that we only deal with a phase-space of functions continuous in time, allowing this general setting, which might be more useful when less regularity is known in the hereditary mechanism. After that, we obtain uniform estimates and asymptotic compactness properties (via an energy method) that allow us to ensure the existence of pullback attractors for the associated process to the problem. Actually, we obtain two different families of minimal pullback attractors, namely, those of fixed bounded sets but also for a class of time-dependent families (universe) given by a tempered condition. Finally, from comparison results, we establish relations among them, and under suitable additional assumptions we conclude that these families of attractors are in fact the same object.