# American Institute of Mathematical Sciences

September  2016, 15(5): 1809-1823. doi: 10.3934/cpaa.2016016

## 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, Taiwan 3 Department of Applied Physics, Waseda University, Tokyo, 169-8555

Received  October 2015 Revised  May 2016 Published  July 2016

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.
Citation: Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. On small data scattering of Hartree equations with short-range interaction. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1809-1823. doi: 10.3934/cpaa.2016016
##### References:
 [1] Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity,, \emph{Funkacialaj Ekvacioj}, 56 (2013), 193. doi: 10.1619/fesi.56.193. [2] Y. Cho and T. Ozawa, On the semi-relativisitc Hartree type equation,, \emph{SIAM J. Math. Anal.}, 38 (2006), 1060. doi: 10.1137/060653688. [3] Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 1121. doi: 10.3934/cpaa.2011.10.1121. [4] M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space,, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 917. doi: 10.1016/j.anihpc.2008.04.002. [5] N. Hayashi and P. I. Naumkin, Remarks on Scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential,, \emph{SUT J. Math.}, 34 (1998), 13. [6] N. Hayashi and P. I. Naumkin, Scattering theory and asymptotics for large time of solutions to the Hartree type equations with a long range potential,, \emph{Hokkaido Math. J.}, 30 (2001), 137. doi: 10.14492/hokmj/1350911928. [7] N. Hayashi, P. I. Naumkin and T. Ogawa, Scattering operator for semirelativistic Hartree type equation with a short range potential,, \emph{Diff. Int. Equations}, 28 (2015), 1085. [8] N. Hayashi, P.I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation,, \emph{SIAM J. Math. Anal.}, 29 (1998), 1256. doi: 10.1137/S0036141096312222. [9] N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations,, \emph{Ann. Inst. H. Poincare Phys. Theor.}, 46 (1987), 187. [10] S. Herr and T. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity,, \emph{J. Differential Equations}, 259 (2015), 5510. doi: 10.1016/j.jde.2015.06.037. [11] Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces,, \emph{Commun. Pure Appl. Anal.}, 14 (2015), 2265. doi: 10.3934/cpaa.2015.14.2265. [12] H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves,, Oberwolfach Seminars, 45 (2014). [13] K. Nakanishi and T. Ozawa, Scattering Problem for Nonlinear Schrodinger and Hartree Equations,, Tosio Kato's method and principle for evolution equations in mathematical physics (Sapporo, (2001). [14] F. Pusateri, Modified scattering for the Boson star equation,, \emph{Commun. Math. Phys.}, 332 (2014), 1203. doi: 10.1007/s00220-014-2094-x.

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##### References:
 [1] Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity,, \emph{Funkacialaj Ekvacioj}, 56 (2013), 193. doi: 10.1619/fesi.56.193. [2] Y. Cho and T. Ozawa, On the semi-relativisitc Hartree type equation,, \emph{SIAM J. Math. Anal.}, 38 (2006), 1060. doi: 10.1137/060653688. [3] Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 1121. doi: 10.3934/cpaa.2011.10.1121. [4] M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space,, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 917. doi: 10.1016/j.anihpc.2008.04.002. [5] N. Hayashi and P. I. Naumkin, Remarks on Scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential,, \emph{SUT J. Math.}, 34 (1998), 13. [6] N. Hayashi and P. I. Naumkin, Scattering theory and asymptotics for large time of solutions to the Hartree type equations with a long range potential,, \emph{Hokkaido Math. J.}, 30 (2001), 137. doi: 10.14492/hokmj/1350911928. [7] N. Hayashi, P. I. Naumkin and T. Ogawa, Scattering operator for semirelativistic Hartree type equation with a short range potential,, \emph{Diff. Int. Equations}, 28 (2015), 1085. [8] N. Hayashi, P.I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation,, \emph{SIAM J. Math. Anal.}, 29 (1998), 1256. doi: 10.1137/S0036141096312222. [9] N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations,, \emph{Ann. Inst. H. Poincare Phys. Theor.}, 46 (1987), 187. [10] S. Herr and T. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity,, \emph{J. Differential Equations}, 259 (2015), 5510. doi: 10.1016/j.jde.2015.06.037. [11] Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces,, \emph{Commun. Pure Appl. Anal.}, 14 (2015), 2265. doi: 10.3934/cpaa.2015.14.2265. [12] H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves,, Oberwolfach Seminars, 45 (2014). [13] K. Nakanishi and T. Ozawa, Scattering Problem for Nonlinear Schrodinger and Hartree Equations,, Tosio Kato's method and principle for evolution equations in mathematical physics (Sapporo, (2001). [14] F. Pusateri, Modified scattering for the Boson star equation,, \emph{Commun. Math. Phys.}, 332 (2014), 1203. doi: 10.1007/s00220-014-2094-x.
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