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March  2008, 1(1): 71-82. doi: 10.3934/dcdss.2008.1.71

On radial solutions of semi-relativistic Hartree equations

Yonggeun Cho 1, and  Tohru Ozawa 2,

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

Received  July 2006 Revised  August 2007 Published  December 2007

We consider the semi-relativistic Hartree type equation with nonlocal nonlinearity $F(u) = \lambda (|x|^{-\gamma} * |u|^2)u, 0 < \gamma < n, n \ge 1$. In [2, 3], the global well-posedness (GWP) was shown for the value of $\gamma \in (0, \frac{2n}{n+1}), n \ge 2$ with large data and $\gamma \in (2, n), n \ge 3$ with small data. In this paper, we extend the previous GWP result to the case for $\gamma \in (1, \frac{2n-1}n), n \ge 2$ with radially symmetric large data. Solutions in a weighted Sobolev space are also studied.
Keywords: radially symmetric solution., global well-posedness, semi-relativistic Hartree type equation.
Mathematics Subject Classification: Primary: 35Q40, 35Q55; Secondary: 47J3.
Citation: Yonggeun Cho, Tohru Ozawa. On radial solutions of semi-relativistic Hartree equations. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 71-82. doi: 10.3934/dcdss.2008.1.71
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