On small data scattering of Hartree equations with short-range interaction

Yonggeun Cho; Gyeongha  Hwang; Tohru Ozawa

doi:10.3934/cpaa.2016016

Article Contents

2016, Volume 15, Issue 5: 1809-1823. Doi: 10.3934/cpaa.2016016

This issue Previous Article A direct method of moving planes for fractional Laplacian equations in the unit ball Next Article Liouville type theorems for singular integral equations and integral systems

On small data scattering of Hartree equations with short-range interaction

1.
Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756
2.
National Center for Theoretical Sciences, No. 1 Sec. 4 Roosevelt Rd., National Taiwan University, Taipei, 106
3.
Department of Applied Physics, Waseda University, Tokyo, 169-8555

Received: October 2015

Revised: May 2016

Published: September 2016

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Abstract

In this note we study Hartree type equations with $|\nabla|^\alpha (1 < \alpha \le 2)$ and potential whose Fourier transform behaves like $|\xi|^{-(d-\gamma_1)}$ at the origin and $|\xi|^{-(d-\gamma_2)}$ at infinity. We show non-existence of scattering when $0 < \gamma_1 \le 1$ and small data scattering in $H^s$ for $s > \frac{2-\alpha}2$ when $2 < \gamma_1 \le d$ and $0 < \gamma_2 \le 2$. For this we use $U^p-V^p$ space argument and Strichartz estimates.

Keywords:

Mathematics Subject Classification: Primary: M35Q55; Secondary: 35Q40.

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References

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