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

Abstract

We study the Cauchy problem for the fractional Schrödinger equation $$i{\partial }_{t}u=(m^2-\Delta {)}^{\frac{\alpha }{2}}u+F(u)in{\mathbb{R}}^{1+n},$$ where $n\ge 1$, $m\ge 0$, $1<\alpha <2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u)=\lambda (\frac{\psi (\cdot )}{|\cdot {|}^{\gamma }}\ast |u{|}^2)u$$ with $\lambda =\pm 1,0<\gamma <n$, 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 $\lambda =-1$.

References in corpus (1)

• Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation

Cited by in corpus (2)

• Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity
• On the orbital stability of fractional Schrödinger equations