Well-Posedness and Blow-Up for the Fractional Schrödinger- Choquard Equation

Abstract

. In this paper, we study the well-posedness and blow-up solutions for the fractional Schr¨odinger equation with a Hartree-type nonlinearity together with a power-type subcritical or critical perturbations. For nonradial initial data or radial initial data, we prove the local well-posedness for the defocusing and the focusing cases with sub-critical or critical nonlinearity. We obtain the global well-posedness for the defocusing case, and for the focusing mass-subcritical case or mass-critical case with initial data small enough. We also investigate blow-up solutions for the focusing mass-critical problem.