American Institute of Mathematical Sciences

Advanced
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 - A, 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

[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

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

[1]

Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905
[2]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261
[3]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37
[4]

Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010
[5]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023
[6]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[7]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
[8]

Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091
[9]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181
[10]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389
[11]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027
[12]

Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093
[13]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15
[14]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273
[15]

Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100
[16]

Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global well-posedness and scattering for Skyrme wave maps. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1923-1933. doi: 10.3934/cpaa.2012.11.1923
[17]

Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066
[18]

Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934/cpaa.2007.6.997
[19]

Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144
[20]

Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (66)
  • HTML views (111)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences