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

Source abstract

Abstract 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 $\mathbb{R}^{d}$ with $1\leqslant d\leqslant 3$. 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 $\{\mathfrak{A}(\tau )\}_{\tau \in \mathbb{R}}$ in $\mathbb{H}:= H^{1}(\mathbb{R}^{d})\times H^{1}(\mathbb{R}^{d})\times L^{2}(\mathbb{R}^{d})$. When the time-dependent external forces converge to given time-independent functions, we prove that the time-component $\mathfrak{A}(\tau )$ converges to the global attractor (obtained by Guo and Li [21], and Lu and Wang [38]) of the autonomous KGS equation as $\tau$ 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 $[\tau ,+\infty )$ and $(-\infty ,\tau ]$. 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 $\mathbb{H}$, where the difficulties caused by the non-compactness of Sobolev embeddings on $\mathbb{R}^{d}$ and the weak dissipativeness of the KGS equations are surmounted.