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On global solution to the Klein-Gordon-Hartree equation below energy space

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Abstract

In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in R 3. Making use of Bourgain’s method in conjunction with precise Strichartz estimates of S.Klainerman and D.Tataru, we establish the H s-global well-posedness with s < 1 of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation, inspired by I. Gallagher and F. Planchon [6]. As a preceded result, we obtain global solution for this non-scaling equation with small initial datum in Hs0 × Hs0−1 where s0 = γ+6 8γ−6. In doing so a number of nonlinear a prior estimates is established by using Bony’s paraproduct decomposition, flexibility of Klein-Gordon admissible pairs which are a bit different from wave’s and second microlocal estimates in frame of the mixed Besov space. As far as we know, it seems that this is the first result on low regularity for this Klein-Gordon-Hartree equation

Topics: φ|t=0 = φ0 ∂tφ|t=0 = φ1
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