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

Abstract

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.