### On the modified scattering of $3$-d Hartree type fractional Schrödinger equations with Coulomb potential for any given initial and boundary data.

#### Abstract

In this paper, we study 3-d Hartree type fractional Schrödinger equations: $$i\partial_{t}u-\vert\nabla\vert^{\alpha}u = \lambda \left ( |x|^{-\gamma} *\vert u\vert^{2} \right ) u,\;\;1 < \alpha < 2,\;\;0 < \gamma < 3,\; \lambda \in \mathbb R \setminus \{0\}.$$ In [7] it is known that no scattering occurs in $L^2$ for the long range ($0 < \gamma \le 1$). In [4, 10, 8] the short-range scattering ($1 < \gamma < 3$) was treated for the scattering in $H^s$. In this paper, we consider the critical case ($\gamma = 1$) and prove a modified scattering in $L^\infty$ on the frequency to the Cauchy problem with small initial data. For this purpose, we investigate the global behavior of $x e^{it |\nabla|^\alpha } u$, $x^2 e^{it |\nabla|^\alpha } u$ and $\langle\xi\rangle ^5 \widehat{e^{it |\nabla|^\alpha } u}$. Due to the non-smoothness of $|\nabla|^\alpha$ near zero frequency the range of $\alpha$ is restricted to $(\frac{17}{10}, 2)$.

#### Article information

Source
Adv. Differential Equations, Volume 23, Number 9/10 (2018), 649-692.

Dates
First available in Project Euclid: 13 June 2018

Cho, Yonggeun; Hwang, Gyeongha; Yang, Changhun. On the modified scattering of $3$-d Hartree type fractional Schrödinger equations with Coulomb potential for any given initial and boundary data. Adv. Differential Equations 23 (2018), no. 9/10, 649--692. https://projecteuclid.org/euclid.ade/1528855474