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

By Yonggeun Cho, Gyeongha Hwang, Hichem Hajaiej, Tohru Ozawa. 2012

129 citations recorded by the source corpus.

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 $\lambda = -1$.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fd973ad3c84337b98326a096d8ca5c01ce7060e0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

On Fractional Schrodinger Equations in sobolev spaces (2015, background): “When the nonlinearity is Hartree-type such as pV ̊ |u|p ́1qu, some amount of loss of derivatives can be absorbed by the regularizing property of the convoluted potential, depending on how nice the potential is (see [6] and references there in).”
Sharp threshold of blow-up and scattering for the fractional Hartree equation (2017, methodology): “To the authors’ knowledge, Cho et al in [8] gave some small data results.”