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A unified approach for energy scattering for focusing nonlinear Schrödinger equations
1. | Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France |
2. | Department of Mathematics, HCMC University of Education, 280 An Duong Vuong, Ho Chi Minh, Vietnam |
$ i\partial_t u + \Delta u = - |u|^\alpha u, \quad (t, x) \in \mathbb R \times \mathbb R^N, $ |
$ N\geq 1 $ |
$ \alpha>\frac{4}{N} $ |
$ \alpha <\frac{4}{N-2} $ |
$ N\geq 3 $ |
References:
[1] |
T. Akahori and H. Nawa,
Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672.
doi: 10.1215/21562261-2265914. |
[2] |
A. K. Arora, B. Dodson and J. Murphy,
Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc., 148 (2020), 1653-1663.
doi: 10.1090/proc/14824. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[4] |
T. Cazenave and F. B. Weissler,
Rapidly decaying solutions of the nonlinear Schrödinger equation, Commun. Math. Phys., 147 (1992), 75-100.
doi: 10.1007/BF02099529. |
[5] |
B. Dodson and J. Murphy,
A new proof of scattering below the ground state for the non-radial focusing NLS, Math. Res. Lett., 25 (2018), 1805-1825.
doi: 10.4310/MRL.2018.v25.n6.a5. |
[6] |
T. Duyckaerts, J. Holmer and S. Roudenko,
Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.
doi: 10.4310/MRL.2008.v15.n6.a13. |
[7] |
T. Duyckaerts and S. Roudenko,
Threshold solutions for the focusing 3D cubic Schrödinger equation, Rev. Mat. Iberoam., 26 (2010), 1-56.
doi: 10.4171/RMI/592. |
[8] |
T. Duyckaerts and S. Roudenko,
Going beyond the threshold: Scattering and blpw-up in the focusing NLS equation, Commun. Math. Phys., 334 (2015), 1573-1615.
doi: 10.1007/s00220-014-2202-y. |
[9] |
D. Fang, J. Xie and T. Cazenave,
Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.
doi: 10.1007/s11425-011-4283-9. |
[10] |
Y. Gao and Z. Wang, Below and beyond the mass-energy threshold: Scattering for the Hartree equation with radial data in $d\geq 5$, Z. Angew. Math. Phys., 71 (2020), 23pp.
doi: 10.1007/s00033-020-1274-0. |
[11] |
C. D. Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, 2014, 177–243.
doi: 10.1002/cta.2381. |
[12] |
J. Holmer and S. Roudenko,
A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.
doi: 10.1007/s00220-008-0529-y. |
[13] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[14] |
C. E. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[15] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[16] |
S. Xia and C. Xu,
On dynamics of the system of two coupled nonlinear Schrödinger in $ \mathbb R^3$, Math. Meth. Appl. Sci., 42 (2019), 7096-7112.
doi: 10.1002/mma.5814. |
show all references
References:
[1] |
T. Akahori and H. Nawa,
Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672.
doi: 10.1215/21562261-2265914. |
[2] |
A. K. Arora, B. Dodson and J. Murphy,
Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc., 148 (2020), 1653-1663.
doi: 10.1090/proc/14824. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[4] |
T. Cazenave and F. B. Weissler,
Rapidly decaying solutions of the nonlinear Schrödinger equation, Commun. Math. Phys., 147 (1992), 75-100.
doi: 10.1007/BF02099529. |
[5] |
B. Dodson and J. Murphy,
A new proof of scattering below the ground state for the non-radial focusing NLS, Math. Res. Lett., 25 (2018), 1805-1825.
doi: 10.4310/MRL.2018.v25.n6.a5. |
[6] |
T. Duyckaerts, J. Holmer and S. Roudenko,
Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.
doi: 10.4310/MRL.2008.v15.n6.a13. |
[7] |
T. Duyckaerts and S. Roudenko,
Threshold solutions for the focusing 3D cubic Schrödinger equation, Rev. Mat. Iberoam., 26 (2010), 1-56.
doi: 10.4171/RMI/592. |
[8] |
T. Duyckaerts and S. Roudenko,
Going beyond the threshold: Scattering and blpw-up in the focusing NLS equation, Commun. Math. Phys., 334 (2015), 1573-1615.
doi: 10.1007/s00220-014-2202-y. |
[9] |
D. Fang, J. Xie and T. Cazenave,
Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.
doi: 10.1007/s11425-011-4283-9. |
[10] |
Y. Gao and Z. Wang, Below and beyond the mass-energy threshold: Scattering for the Hartree equation with radial data in $d\geq 5$, Z. Angew. Math. Phys., 71 (2020), 23pp.
doi: 10.1007/s00033-020-1274-0. |
[11] |
C. D. Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, 2014, 177–243.
doi: 10.1002/cta.2381. |
[12] |
J. Holmer and S. Roudenko,
A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.
doi: 10.1007/s00220-008-0529-y. |
[13] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[14] |
C. E. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[15] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[16] |
S. Xia and C. Xu,
On dynamics of the system of two coupled nonlinear Schrödinger in $ \mathbb R^3$, Math. Meth. Appl. Sci., 42 (2019), 7096-7112.
doi: 10.1002/mma.5814. |
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