Abstract. We study the Cauchy problem for the fractional Schrö dinger equation, where n b 1, m b 0, 1 < a < 2, and F stands for the nonlinearity of Hartree type F ðuÞ ¼ lðcðÁÞj Á j Àg à juj 2 Þu with l ¼G1, 0 < g < n,and 0 a c A L y ðR n Þ. We prove the existence and uniqueness of local and global solutions for certain a, g, l, c. We also remark on finite time blowup of solutions when l ¼ À1.

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We consider the fractional Schrödinger equations with focusing Hartree type nonlinearities. When the energy is negative, we show that the solution blows up in a finite time. For this purpose, based on Glassey's argument, we obtain a virial type inequality.2010 Mathematics Subject Classification. M35Q55, 35Q40.

We study, under the radial symmetry assumption, the solutions to the fractional Schrödinger equations of critical nonlinearity in R 1+d , d ≥ 2, with Lévy index 2d/(2d − 1) < α < 2. We firstly prove the linear profile decomposition and then apply it to investigate the properties of the blowup solutions of the nonlinear equations with mass-critical Hartree type nonlineartity.2010 Mathematics Subject Classification. 35Q55, 35Q40.

We show the existence of ground state and orbital stability of standing waves of fractional Schrödinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.

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In this note we study Hartree type equations with |∇| α (1 < α ≤ 2) and potential whose Fourier transform behaves like |ξ| −(d−γ 1) at the origin and |ξ| −(d−γ 2) at infinity. We show non-existence of scattering when 0 < γ 1 ≤ 1 and small data scattering in H s for s > 2−α 2 when 2 < γ 1 ≤ d and 0 < γ 2 ≤ 2. For this we use U p − V p space argument and Strichartz estimates.