Abstract
We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with fractional Laplacian in the Schrödinger equation in R1+1. We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schrödinger data in H s1 and wave data in H s2 ×H s2−1 for 3/4−α < s 1 ≤ 0 and −1/2 < s 2 < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α ≤ 1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s 1 = 0 and −1/2 < s 2 < 1/2 by using the conservation law for the L 2 norm of u.
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Huang, C., Guo, B., Huang, D. et al. Global well-posedness of the fractional Klein-Gordon-Schrödinger system with rough initial data. Sci. China Math. 59, 1345–1366 (2016). https://doi.org/10.1007/s11425-016-5133-6
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DOI: https://doi.org/10.1007/s11425-016-5133-6