Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain

We study the backwards dynamics for the wave equation defined on the whole 3D Euclid space with a positively bounded coefficient of the damping and a time-dependent force. We introduce a backwards compact attractor which is the minimal one among the backwards compact and pullback attracting sets. We prove that a backwards compact attractor is equivalent to a pullback attractor (invariant) that is backwards compact, i.e. the union of the attractor over the past time is pre-compact. We also establish a sufficient and necessary criterion of the existence of a backwards compact attractor and show the relationship of a periodic attractor. As an application of these abstract results, we prove that the non autonomous wave equation has a backwards compact attractor under some backwards assumptions of the non-autonomous force. Moreover, we establish the backwards compactness from some periodicity assumptions, more precisely, if the force is assumed only to be periodic then a backwards compact attractor exists, and if the damped coefficient is further assumed to be periodic then the attractor is both periodic and backwards compact. (C) 2017 Elsevier Ltd. All rights reserved.