Pullback exponential attractors with admissible exponential growth in the past

For an evolution process we prove the existence of a pullback exponential attractor, a positively invariant family of compact subsets which have a uniformly bounded fractal dimension and pullback attract all bounded subsets at an exponential rate. The construction admits the exponential growth in the past of the sets forming the family and generalizes the known approaches. It also allows to substitute the smoothing property by a weaker requirement without auxiliary spaces. The theory is illustrated with the examples of a nonautonomous Chafee–Infante equation and a time-dependent perturbation of a reaction–diffusion equation improving the results known in the literature.