Remarks on the semirelativistic Hartree equations

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October 2009, 23(4): 1277-1294. doi: 10.3934/dcds.2009.23.1277

Remarks on the semirelativistic Hartree equations

Yonggeun Cho ^1, , Tohru Ozawa ^2, , Hironobu Sasaki ^3, and Yongsun Shim ^4,

1.	Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756, South Korea
2.	Department of Mathematics, Hokkaido University, Sapporo 060-0810
3.	Department of Mathematics, Osaka University, Toyonaka 563-0043
4.	Department of Mathematics, POSTECH, Pohang 790-784, South Korea

Received January 2007 Revised August 2007 Published November 2008

Abstract
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We study the global well-posedness (GWP) and small data scattering of radial solutions of the semirelativistic Hartree type equations with nonlocal nonlinearity $F(u) = \lambda (|\cdot|^{-\gamma}$ * $|u|^2)u$, $\lambda \in \mathbb{R} \setminus \{0\}$, $0 < \gamma < n$, $n \ge 3$. We establish a weighted $L^2$ Strichartz estimate applicable to non-radial functions and some fractional integral estimates for radial functions.

Keywords: radial solutions, semirelativistic Hartree type equations, global well-posedness, scattering.

Mathematics Subject Classification: Primary: 35Q40, 35Q55; Secondary: 47J35.

Citation: Yonggeun Cho, Tohru Ozawa, Hironobu Sasaki, Yongsun Shim. Remarks on the semirelativistic Hartree equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1277-1294. doi: 10.3934/dcds.2009.23.1277

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