# 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, Funkacialaj Ekvacioj, 56 (2013), 193-224. doi: 10.1619/fesi.56.193.  Google Scholar [2] Y. Cho and T. Ozawa, On the semi-relativisitc Hartree type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688.  Google Scholar [3] Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.  Google Scholar [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), 917-941. doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar [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, SUT J. Math., 34 (1998), 13-24.  Google Scholar [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, Hokkaido Math. J., 30 (2001), 137-161. doi: 10.14492/hokmj/1350911928.  Google Scholar [7] N. Hayashi, P. I. Naumkin and T. Ogawa, Scattering operator for semirelativistic Hartree type equation with a short range potential, Diff. Int. Equations, 28 (2015), 1085-1104.  Google Scholar [8] N. Hayashi, P.I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation, SIAM J. Math. Anal., 29 (1998), 1256-1267. doi: 10.1137/S0036141096312222.  Google Scholar [9] N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincare Phys. Theor., 46 (1987), 187-213.  Google Scholar [10] S. Herr and T. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity, J. Differential Equations, 259 (2015), 5510-5532. doi: 10.1016/j.jde.2015.06.037.  Google Scholar [11] Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282. doi: 10.3934/cpaa.2015.14.2265.  Google Scholar [12] H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, 45 2014. Google Scholar [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).  Google Scholar [14] F. Pusateri, Modified scattering for the Boson star equation, Commun. Math. Phys., 332 (2014), 1203-1234. doi: 10.1007/s00220-014-2094-x.  Google Scholar

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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, Funkacialaj Ekvacioj, 56 (2013), 193-224. doi: 10.1619/fesi.56.193.  Google Scholar [2] Y. Cho and T. Ozawa, On the semi-relativisitc Hartree type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688.  Google Scholar [3] Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.  Google Scholar [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), 917-941. doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar [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, SUT J. Math., 34 (1998), 13-24.  Google Scholar [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, Hokkaido Math. J., 30 (2001), 137-161. doi: 10.14492/hokmj/1350911928.  Google Scholar [7] N. Hayashi, P. I. Naumkin and T. Ogawa, Scattering operator for semirelativistic Hartree type equation with a short range potential, Diff. Int. Equations, 28 (2015), 1085-1104.  Google Scholar [8] N. Hayashi, P.I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation, SIAM J. Math. Anal., 29 (1998), 1256-1267. doi: 10.1137/S0036141096312222.  Google Scholar [9] N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincare Phys. Theor., 46 (1987), 187-213.  Google Scholar [10] S. Herr and T. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity, J. Differential Equations, 259 (2015), 5510-5532. doi: 10.1016/j.jde.2015.06.037.  Google Scholar [11] Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282. doi: 10.3934/cpaa.2015.14.2265.  Google Scholar [12] H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, 45 2014. Google Scholar [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).  Google Scholar [14] F. Pusateri, Modified scattering for the Boson star equation, Commun. Math. Phys., 332 (2014), 1203-1234. doi: 10.1007/s00220-014-2094-x.  Google Scholar
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