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Local smooth solutions of the nonlinear Klein-gordon equation
The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data
1. | Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France |
2. | Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam |
Based on recent works of Dodson-Murphy [
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,
Scattering of radial data in the focusing NLS and generalized Hartree equations, Discrete Cont. Dyn. Syst., 39 (2019), 6643-6668.
doi: 10.3934/dcds.2019289. |
[3] |
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. |
[4] |
A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, preprint, arXiv: 1904.05339. Google Scholar |
[5] |
L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, preprint, arXiv: 1905.02663. Google Scholar |
[6] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, AMS, 2003.
doi: 10.1090/cln/010. |
[7] |
T. Cazenave, D. Fang and J. Xie,
Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.
doi: 10.1007/s11425-011-4283-9. |
[8] |
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. |
[9] |
C. V. Coffman,
Uniqueness of the ground state solution for $\Delta u - u + u^3 = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.
doi: 10.1007/BF00250684. |
[10] |
V. D. Dinh, Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions, preprint, arXiv: 1908.02987. Google Scholar |
[11] |
V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order Schrödinger equations, preprint, arXiv: 2001.03022. Google Scholar |
[12] |
B. Dodson and J. Murphy,
A new proof of scattering below the ground state for the 3D radial focusing NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867.
doi: 10.1090/proc/13678. |
[13] |
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. |
[14] |
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. |
[15] |
L. G. Farah and C. Guzman,
Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation, J. Differential Equations, 262 (2017), 4175-4231.
doi: 10.1016/j.jde.2017.01.013. |
[16] |
L. G. Farah and C. Guzman,
Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc., 51 (2020), 449-512.
doi: 10.1007/s00574-019-00160-1. |
[17] |
D. Foschi,
Inhomogeneous strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.
doi: 10.1142/S0219891605000361. |
[18] |
C. Guevara,
Global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express., 2014 (2014), 177-243.
|
[19] |
Q. Guo,
Scattering for the focusing $L^2$-supercritical and $\dot{H}^2$-subcritical biharmonic NLS equations, Comm. Partial Differential Equations, 41 (2016), 185-207.
doi: 10.1080/03605302.2015.1116556. |
[20] |
Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549. Google Scholar |
[21] |
M. Hamano and M. Ikeda, Global dynamics below the ground state for the focusing Schrödinger equation with a potential, J. Evol. Equ., 2019 (in press). Google Scholar |
[22] |
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. |
[23] |
Y. Hong,
Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal., 15 (2016), 1571-1601.
doi: 10.3934/cpaa.2016003. |
[24] |
M. Keel and T. Tao,
Endpoint strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[25] |
C. E. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[26] |
J. Krieger, E. Lenzmann and P. Raphaël,
On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS, Ann. Henri Poincaré, 10 (2009), 1159-1205.
doi: 10.1007/s00023-009-0010-2. |
[27] |
M. K. Kwong,
Uniqueness of positive solution of $\Delta u - u + u^p = 0$ in $ \mathbb R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[28] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105.
doi: 10.1002/sapm197757293. |
[29] |
C. Miao, G. Xu and L. Zhao,
The cauchy problem of the hartree equation, J. Partial Diff. Eqs., 21 (2008), 22-44.
|
[30] |
V. Moroz and J. Van Schaftingen,
Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[31] |
T. Ogawa and Y. Tsutsumi,
Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
doi: 10.1016/0022-0396(91)90052-B. |
[32] |
C. Sun, H. Wang, X. Yao and J. Zheng,
Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228.
doi: 10.3934/dcds.2018091. |
[33] |
E. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, 1993. |
[34] |
J. Stubbe,
Global solutions and stable ground states of nonlinear Schrödinger equations, Phys. D, 48 (1991), 259-272.
doi: 10.1016/0167-2789(91)90087-P. |
[35] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[36] |
T. Tao,
On the asymptotic behavior of large radial data for a focusing nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), 1-48.
doi: 10.4310/DPDE.2004.v1.n1.a1. |
[37] |
T. Tao, M. Visan and X. Zhang,
The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.
doi: 10.1080/03605300701588805. |
[38] |
M. C. Vilela,
Inhomogeneous strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.
doi: 10.1090/S0002-9947-06-04099-2. |
[39] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.
|
[40] |
C. L. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 134, 25 pp.
doi: 10.1007/s00526-016-1068-6. |
[41] |
C. Xu and T. Zhao, A remark on the scattering theory for the 2D radial focusing INLS, preprint, arXiv: 1908.00743. Google Scholar |
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,
Scattering of radial data in the focusing NLS and generalized Hartree equations, Discrete Cont. Dyn. Syst., 39 (2019), 6643-6668.
doi: 10.3934/dcds.2019289. |
[3] |
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. |
[4] |
A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, preprint, arXiv: 1904.05339. Google Scholar |
[5] |
L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, preprint, arXiv: 1905.02663. Google Scholar |
[6] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, AMS, 2003.
doi: 10.1090/cln/010. |
[7] |
T. Cazenave, D. Fang and J. Xie,
Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.
doi: 10.1007/s11425-011-4283-9. |
[8] |
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. |
[9] |
C. V. Coffman,
Uniqueness of the ground state solution for $\Delta u - u + u^3 = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.
doi: 10.1007/BF00250684. |
[10] |
V. D. Dinh, Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions, preprint, arXiv: 1908.02987. Google Scholar |
[11] |
V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order Schrödinger equations, preprint, arXiv: 2001.03022. Google Scholar |
[12] |
B. Dodson and J. Murphy,
A new proof of scattering below the ground state for the 3D radial focusing NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867.
doi: 10.1090/proc/13678. |
[13] |
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. |
[14] |
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. |
[15] |
L. G. Farah and C. Guzman,
Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation, J. Differential Equations, 262 (2017), 4175-4231.
doi: 10.1016/j.jde.2017.01.013. |
[16] |
L. G. Farah and C. Guzman,
Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc., 51 (2020), 449-512.
doi: 10.1007/s00574-019-00160-1. |
[17] |
D. Foschi,
Inhomogeneous strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.
doi: 10.1142/S0219891605000361. |
[18] |
C. Guevara,
Global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express., 2014 (2014), 177-243.
|
[19] |
Q. Guo,
Scattering for the focusing $L^2$-supercritical and $\dot{H}^2$-subcritical biharmonic NLS equations, Comm. Partial Differential Equations, 41 (2016), 185-207.
doi: 10.1080/03605302.2015.1116556. |
[20] |
Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549. Google Scholar |
[21] |
M. Hamano and M. Ikeda, Global dynamics below the ground state for the focusing Schrödinger equation with a potential, J. Evol. Equ., 2019 (in press). Google Scholar |
[22] |
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. |
[23] |
Y. Hong,
Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal., 15 (2016), 1571-1601.
doi: 10.3934/cpaa.2016003. |
[24] |
M. Keel and T. Tao,
Endpoint strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[25] |
C. E. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[26] |
J. Krieger, E. Lenzmann and P. Raphaël,
On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS, Ann. Henri Poincaré, 10 (2009), 1159-1205.
doi: 10.1007/s00023-009-0010-2. |
[27] |
M. K. Kwong,
Uniqueness of positive solution of $\Delta u - u + u^p = 0$ in $ \mathbb R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[28] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105.
doi: 10.1002/sapm197757293. |
[29] |
C. Miao, G. Xu and L. Zhao,
The cauchy problem of the hartree equation, J. Partial Diff. Eqs., 21 (2008), 22-44.
|
[30] |
V. Moroz and J. Van Schaftingen,
Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[31] |
T. Ogawa and Y. Tsutsumi,
Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
doi: 10.1016/0022-0396(91)90052-B. |
[32] |
C. Sun, H. Wang, X. Yao and J. Zheng,
Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228.
doi: 10.3934/dcds.2018091. |
[33] |
E. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, 1993. |
[34] |
J. Stubbe,
Global solutions and stable ground states of nonlinear Schrödinger equations, Phys. D, 48 (1991), 259-272.
doi: 10.1016/0167-2789(91)90087-P. |
[35] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[36] |
T. Tao,
On the asymptotic behavior of large radial data for a focusing nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), 1-48.
doi: 10.4310/DPDE.2004.v1.n1.a1. |
[37] |
T. Tao, M. Visan and X. Zhang,
The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.
doi: 10.1080/03605300701588805. |
[38] |
M. C. Vilela,
Inhomogeneous strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.
doi: 10.1090/S0002-9947-06-04099-2. |
[39] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.
|
[40] |
C. L. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 134, 25 pp.
doi: 10.1007/s00526-016-1068-6. |
[41] |
C. Xu and T. Zhao, A remark on the scattering theory for the 2D radial focusing INLS, preprint, arXiv: 1908.00743. Google Scholar |
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