On fractional Schrodinger equations with Hartree type nonlinearity

DoctorIn this dissertation we consider for the fractional Schrödinger equationiut = (-Δ)^α/2 u + F(u) in R^1+n, n ≥ 1with the Lévy index 1 < α < 2 and the nonlinearity F(u) = λ(jxj^-ν*juj^2)u0 < ν < n.In Chapter 1 we study the Cauchy problem for the fractional Schröodinger equation. We prove the existence and uniqueness of local and global solutionsfor certain α and ν. We also remark on finite time blowup of solutions whenλ = -1.In Chapter 2 we develop a profile decomposition of fractional Schröodingerequation with Lévvy index 1 < α < 2. One the main difficulty is the non-locality of fractional operator which causes the lack of Galilean invariance.The second one is the regularity loss of Strichartz estimate stemming from thelow index α < 2. To overcome these difficulties we assume radial symmetry and use the recently developed Strichartz estimates. We will apply the profile decomposition to the blowup profile of fractional Hartree equations