On the semi-relativistic Hartree type equation

Preprint Series # 773
Cho, Yonggeun and Ozawa, Tohru On the semi-relativistic Hartree type equation. (2006);

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Abstract

We study the global Cauchy problem and scattering problem for the semi-relativistic equation in $\mathbb{R}^n, n \ge 1$ with nonlocal nonlinearity $F(u) = \lambda (|x|^{-\gamma} * |u|^2)u, 0 <\gamma < n$. We prove the existence and uniqueness of global solutions for $0 < \gamma < \frac{2n}{n+1}, n \ge 2$ or $\gamma > 2, n \ge 3$ and the non-existence of asymptotically free solutions for $0 < \gamma \le 1, n\ge 3$. We also specify asymptotic behavior of solutions as the mass tends to zero and infinity.

Item Type: Preprint 10 semi-relativistic Hartree type equation, global solution, scattering, non-existence of asymptotically free solution 35-xx PARTIAL DIFFERENTIAL EQUATIONS 1511