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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

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 and Continuous Dynamical Systems, 2009, 23 (4) : 1277-1294. doi: 10.3934/dcds.2009.23.1277
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