Pullback Asymptotic Compactness and Asymptotically Autonomous Robustness of Non-autonomous Klein–Gordon–Schrödinger System on Unbounded Domains

We investigate the pullback asymptotic compactness of solution operators and the asymptotically autonomous robustness of pullback attractors for a non-autonomous Yukawa coupling Klein–Gordon–Schrödinger (KGS) equations defined on an unbounded domain Rᵈ with 1 d 3. Under some new high-order integrability conditions found in this paper on the time-dependent external forces, we demonstrate that the non-autonomous dynamical system associated with the solution operator admits a unique pullback attractor (τ )\τ ∈ R in H:= H¹(Rᵈ)× H¹(Rᵈ)× L²(Rᵈ). When the time-dependent external forces converge to given time-independent functions, we prove that the time-component A(τ ) converges to the global attractor (obtained by Guo and Li [21], and Lu and Wang [38]) of the autonomous KGS equation as τ tends to positive and negative infinity, respectively. Unlike the methods of Caraballo et al. [14], Kinra et al. [26], and Wang et al. [58], we give up using the uniform pullback asymptotic compactness of the solution operators over the infinite time-intervals [τ ,+∞ ) and (-∞ ,τ ]. The famous idea of energy equations due to Ball [6] and method of uniform tail-ends estimates of Wang [52], originally used in the autonomous case, are adapted in the nonautonomous case in order to derive the pullback asymptotic compactness of the solution operators in H, where the difficulties caused by the non-compactness of Sobolev embeddings on Rᵈ and the weak dissipativeness of the KGS equations are surmounted.