Autonomous and non-autonomous unbounded attractors under perturbations

Pullback attractors with forwards unbounded behaviour are to be found in the literature, but not much is known about pullback attractors with each and every section being unbounded. In this paper, we introduce the concept of unbounded pullback attractor, for which the sections are not required to be compact. These objects are addressed in this paper in the context of a class of non-autonomous semilinear parabolic equations. The nonlinearities are assumed to be non-dissipative and, in addition, defined in such a way that the equation possesses unbounded solutions as the initial time goes to -infinity, for each elapsed time. Distinct regimes for the non-autonomous term are taken into account. Namely, we address the small non-autonomous perturbation and the asymptotically autonomous cases.