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A direct method of moving planes for fractional Laplacian equations in the unit ball
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincare Phys. Theor., 46 (1987), 187-213. |
[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. |
[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. |
[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, Commun. Math. Phys., 332 (2014), 1203-1234.
doi: 10.1007/s00220-014-2094-x. |
show all references
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincare Phys. Theor., 46 (1987), 187-213. |
[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. |
[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. |
[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, Commun. Math. Phys., 332 (2014), 1203-1234.
doi: 10.1007/s00220-014-2094-x. |
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