The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data

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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

Van Duong Dinh ^1,2, and Sahbi Keraani ^1,,

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.

Keywords: Nonlinear Schrödinger equation, Nonlinear Choquard equation, Ground state, Scattering, Radial Sobolev embedding.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35P25.

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. 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.
[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. 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.
[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. 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.
[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. 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.

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.
[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. 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.
[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. 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.
[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. 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.

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