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November  2020, 40(11): 6441-6471. doi: 10.3934/dcds.2020286

A unified approach for energy scattering for focusing nonlinear Schrödinger equations

Van Duong Dinh 1,2,

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

Received  March 2020 Published  July 2020

We consider the Cauchy problem for focusing nonlinear Schrödinger equation
$ i\partial_t u + \Delta u = - |u|^\alpha u, \quad (t, x) \in \mathbb R \times \mathbb R^N, $
where
$ N\geq 1 $
,
$ \alpha>\frac{4}{N} $
 and
$ \alpha <\frac{4}{N-2} $
 if
$ N\geq 3 $
. We give a criterion for energy scattering for the equation that covers well-known scattering results below, at and above the mass and energy ground state threshold. The proof is based on a recent argument of Dodson-Murphy [Math. Res. Lett. 25(6):1805-1825, 2018] using the interaction Morawetz estimate.
Keywords: Nonlinear Schrödinger equation, ground state, global well-posedness, compactness, scattering.
Mathematics Subject Classification: Primary: 35A01; Secondary: 35Q55.
Citation: Van Duong Dinh. A unified approach for energy scattering for focusing nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6441-6471. doi: 10.3934/dcds.2020286
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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