On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity

Original · EN

We study the Cauchy problem for the fractional Schrödinger equation i∂ₜu = (m²-Δ)α2 u + 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.