August  2021, 14(8): 2837-2876. doi: 10.3934/dcdss.2020407

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

* Corresponding author: Sahbi Keraani

Received  January 2020 Revised  June 2020 Published  August 2021 Early access  July 2020

Based on recent works of Dodson-Murphy [12] and Arora-Dodson-Murphy [3], we give a unified approach for the energy scattering with radially symmetric initial data for nonlinear Schrödinger equations and nonlinear Choquard equations in any dimensions $ N\geq 2 $. We also discuss its applications for other Schrödinger-type equations.

Citation: Van Duong Dinh, Sahbi Keraani. The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2837-2876. doi: 10.3934/dcdss.2020407
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. AroraB. 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.

[5]

L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, preprint, arXiv: 1905.02663.

[6]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, AMS, 2003. doi: 10.1090/cln/010.

[7]

T. CazenaveD. 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.

[11]

V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order Schrödinger equations, preprint, arXiv: 2001.03022.

[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. DuyckaertsJ. 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.

[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).

[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. KriegerE. 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. MiaoG. 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. SunH. WangX. 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. TaoM. 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.

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. AroraB. 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.

[5]

L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, preprint, arXiv: 1905.02663.

[6]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, AMS, 2003. doi: 10.1090/cln/010.

[7]

T. CazenaveD. 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.

[11]

V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order Schrödinger equations, preprint, arXiv: 2001.03022.

[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. DuyckaertsJ. 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.

[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).

[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. KriegerE. 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. MiaoG. 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. SunH. WangX. 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. TaoM. 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.

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