MR3114821 Yonggeun CHO, Hichem HAJAIEJ, Gyeongha HWANG and Tohru OZAWA Yonggeun CHO, Hichem HAJAIEJ, Gyeongha HWANG and Tohru OZAWA On the Cauchy Problem of Fractional Schrödinger Equation with Hartree Type Nonlinearity Funkcialaj Ekvacioj. Serio Internacia 56 2013 193--224 http://fe.math.kobe-u.ac.jp/FE/FullPapers/56-2/56_193.pdf http://www.ams.org/mathscinet-getitem?mr=MR3114821 We study the Cauchy problem for the fractional Schrödinger equation $i\partial_tu=(m^2-\Delta)^{\alpha/2}u+F(u)$ in $\mathbb{R}^{1+n}$, where $n\ge1$, $m\ge0$, $1‹\alpha‹2$, and $F$ stands for the nonlinearity of Hartree type $F(u)=\lambda(\psi(\cdot)|\cdot|^{-\gamma}*|u|^2)u$ with $\lambda=\pm1$, $0‹\gamma‹n$, and $0\le\psi\in L^\infty(\mathbb R^n)$. We prove the existence and uniqueness of local and global solutions for certain $\alpha$, $\gamma$, $\lambda$, $\psi$. We also remark on finite time blowup of solutions when $\lambda=-1$. Fractional Schrödinger equation, Hartree type nonlinearity, Strichartz estimates, Finite time blowup. 35Q40, 35Q55, 47J35. 56-193 2013 On the Cauchy Problem of Fractional Schrödinger Equation with Hartree Type Nonlinearity Yonggeun CHO, Hichem HAJAIEJ, Gyeongha HWANG and Tohru OZAWA Yonggeun CHO, Hichem HAJAIEJ, Gyeongha HWANG and Tohru OZAWA
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