Remarks on the semirelativistic Hartree equations

Yonggeun Cho; Tohru Ozawa; Hironobu Sasaki; Yongsun Shim

doi:10.3934/dcds.2009.23.1277

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2009, Volume 23, Issue 4: 1277-1294. Doi: 10.3934/dcds.2009.23.1277

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Remarks on the semirelativistic Hartree equations

1.
Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756
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

Received: December 31, 2006

Revised: July 31, 2007

Published: October 2009

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Abstract

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:

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

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

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