Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations
The authors give a necessary and sufficient condition for the existence of pullback-$\mathcal{D}$ attractors for processes defined on a Banach space of infinite sequences. The result generalizes the results of \textit{X.-Q. Zhao} and \textit{S. Zhou} [ibid. 9, No. 3--4, 763--785 (2008; Zbl 1149.37040)]. The setup is suitable for analyzing lattice dynamical systems. An application of the main result is given to a discrete system associated with the non-autonomous Klein-Gordon-Schrödinger system, an important set of partial differential equations in quantum field theory. The discrete system studied here was also examined by \textit{C. Zhao} and \textit{S. Zhou} [J. Math. Anal. Appl. 332, No. 1, 32--56 (2007; Zbl 1113.37057)] whereby, under certain assumptions on the non-autonomous source terms, those authors prove compact kernel sections for the discrete equations (pullback-$\mathcal{D}$ attractors have a more general basin of attraction than kernel sections). Based on the work in [\textit{G. Łukaszewicz} and \textit{J. C. Robinson}, Discrete Contin. Dyn. Syst. 34, No. 10, 4211--4222 (2014; Zbl 1351.37076)], the final result shows that the pullback-$\mathcal{D}$ attractor associated with the discretized system admits a family of invariant Borel probability measures. Such measures are useful in modeling turbulence.
scientific article; zbMATH DE number 6996864
scientific article