Electron. J. Differential Equations, Vol. 2021 (2021), No. 101, pp. 1-23.

Global well-posedness for Klein-Gordon-Hartree and fractional Hartree equations on modulation spaces

Divyang G. Bhimani

We study the Cauchy problems for the Klein-Gordon (HNLKG), wave (HNLW), and Schrodinger (HNLS) equations with cubic convolution (of Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrodinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. As a consequence, we get local and global well-posedness and scattering in larger than usual $L^p$ -Sobolev spaces.

Submitted April 30, 2021. Published December 21, 2021.
Math Subject Classifications: 35L71, 35Q55, 42B35, 35A01.
Key Words: Klein-Gordon-Hartree equation; fractional Hartree equation; wave-Hartree equation; well-posedness; modulation spaces; small initial data.

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Divyang G. Bhimani
Department of Mathematics
Indian Institute of Science Education and Research
Dr. Homi Bhabha Road, Pune 411008, India
email: divyang.bhimani@iiserpune.ac.in

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