Global attractors for the discrete Klein–Gordon–Schrödinger type equations

Abstract

The purpose of this work is to investigate the asymptotic behaviours of solutions for the discrete Klein–Gordon–Schrödinger type equations in one-dimensional lattice. We first establish the global existence and uniqueness of solutions for the corresponding Cauchy problem. According to the solution's estimate, it is shown that the semi-group generated by the solution is continuous and possesses an absorbing set. Using truncation technique, we show that there exists a global attractor for the semi-group. Finally, we extend the criteria of Zhou et al. [S. Zhou, C. Zhao, and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst. A 21 (2008), pp. 1259–1277.] for finite fractal dimension of a family of compact subsets in a Hilbert space to obtain an upper bound of fractal dimension for the global attractor.

Keywords::

• absorbing set
• global attractor
• truncation technique
• asymptotic nullness
• fractal dimension

Keywords::

• 35B40
• 35Q35
• 37L50

Acknowledgements

Part of this article was finished during Prof. C. Zhao's visiting at the Department of Mathematics, NCU in Taiwan. Prof. C. Zhao thanks for all, especially Prof. Cheng-Hsiung Hsu and Dr Jian-Jhong Lin, their support and hospitality. C.H. Hsu was partially supported by NSC and NCTS of Taiwan. J.J. Lin was partially supported by NSC of Taiwan. C. Zhao partially supported by NSFC of China [grand number 11271290].

Notes

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