On the Cauchy problem of fractional Schr\"{o}dinger equation with Hartree type nonlinearity

TL;DR

This paper investigates the well-posedness and blowup phenomena of solutions to a fractional Schrödinger equation with Hartree nonlinearity, establishing existence, uniqueness, and conditions for finite time blowup.

Contribution

It provides new results on existence, uniqueness, and blowup for the fractional Schrödinger equation with Hartree type nonlinearity, extending previous understanding to fractional orders and nonlinearities.

Findings

Existence and uniqueness of local and global solutions under certain conditions.

Finite time blowup occurs when the nonlinearity is focusing (λ = -1).

Results apply for a range of fractional orders and nonlinear parameters.

Abstract

We study the Cauchy problem for the fractional Schr\"{o}dinger equation $$ i\partial_tu = (m^2-\Delta)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < \alpha < 2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u) = \lambda (\frac{\psi(\cdot)}{|\cdot|^\gamma} * |u|^2)u$$ with $\lambda = \pm1, 0 <\gamma < n$, and $0 \le \psi \in L^\infty(\mathbb R^n)$. We prove the existence and uniqueness of local and global solutions for certain $\alpha$, $\gamma$, $\lambda$, $\psi$. We also remark on finite time blowup of solutions when $\lambda = -1$.