On the Cauchy Problem of Fractional Schrödinger Equation with Hartree Type Nonlinearity

Abstract

We study the Cauchy problem for the fractional Schrödinger equation i∂_tu = (m²−Δ)^α/2u + F(u) in R¹⁺ⁿ, where n ≥ 1, m ≥ 0, 1 < α < 2, and F stands for the nonlinearity of Hartree type F(u) = λ (ψ (·) |·|^−γ ∗ |u|²)u with λ = ±1, 0 < γ < n, and 0 ≤ ψ ∈ L^∞ (Rⁿ). We prove the existence and uniqueness of local and global solutions for certain α, γ, λ, ψ. We also remark on finite time blowup of solutions when λ = −1.