# American Institute of Mathematical Sciences

November  2019, 39(11): 6643-6668. doi: 10.3934/dcds.2019289

## Scattering of radial data in the focusing NLS and generalized Hartree equations

 Department of Mathematics & Statistics, Florida International University, Miami, FL 33199, USA

Received  March 2019 Revised  May 2019 Published  August 2019

We consider the focusing nonlinear Schrödinger equation $i u_t + \Delta u + |u|^{p-1}u = 0$, $p>1,$ and the generalized Hartree equation $iv_t + \Delta v + (|x|^{-(N-\gamma)}\ast |v|^p)|v|^{p-2}u = 0$, $p\geq2$, $\gamma<N$, in the mass-supercritical and energy-subcritical setting. With the initial data $u_0\in H^1( \mathbb R^N)$ the characterization of solutions behavior under the mass-energy threshold is known for the NLS case from the works of Holmer and Roudenko in the radial [15] and Duyckaerts, Holmer and Roudenko in the nonradial setting [10] and further generalizations (see [1,11,13]); for the generalized Hartree case it is developed in [2]. In particular, scattering is proved following the road map developed by Kenig and Merle [16], using the concentration compactness and rigidity approach, which is now standard in the dispersive problems.

In this work we give an alternative proof of scattering for both NLS and gHartree equations in the radial setting in the inter-critical regime, following the approach of Dodson and Murphy [8] for the focusing 3d cubic NLS equation, which relies on the scattering criterion of Tao [26], combined with the radial Sobolev and Morawetz-type estimates. We first generalize it in the NLS case, and then extend it to the nonlocal Hartree-type potential. This method provides a simplified way to prove scattering, which may be useful in other contexts.

Citation: Anudeep Kumar Arora. Scattering of radial data in the focusing NLS and generalized Hartree equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6643-6668. doi: 10.3934/dcds.2019289
##### 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 and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equations, preprint, arXiv: 1904.05339.Google Scholar [3] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. Google Scholar [4] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅱ. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556. 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, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. Google Scholar [7] J. Colliander and S. Roudenko, Mass concentration window size and Strichartz norm divergence rate for the L2-critical nonlinear Schrödinger equation, J. Hyperbolic Differ. Equ., 4 (2007), 613-627. doi: 10.1142/S0219891607001288. Google Scholar [8] B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867. doi: 10.1090/proc/13678. Google Scholar [9] 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 [10] 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 [11] 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 [12] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136. doi: 10.1007/BF01214768. Google Scholar [13] C. D. Guevara, Global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, 2 (2014), 177-243. Google Scholar [14] J. Holmer and S. Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, (2007), Art. ID abm004, 31pp. Google Scholar [15] 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 [16] 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 [17] J. Krieger, E. Lenzmann and P. Raphaël, On stability of pseudo-conformal blowup for L2-critical Hartree NLS, Ann. Henri Poincaré, 10 (2009), 1159-1205. doi: 10.1007/s00023-009-0010-2. Google Scholar [18] M. K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p = 0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar [19] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105. doi: 10.1002/sapm197757293. Google Scholar [20] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 18 (1983), 349–374. doi: 10.2307/2007032. Google Scholar [21] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. Google Scholar [22] P.-L. Lions, Compactness and topological methods for some nonlinear variational problems of mathematical physics, In Nonlinear Problems: Present and Future, (Los Alamos, N.M., 1981), North-Holland Math. Stud., 61, North-Holland, Amsterdam-New York, (1982), 17–34. Google Scholar [23] 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. Google Scholar [24] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813. doi: 10.1007/s11784-016-0373-1. Google Scholar [25] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517. Google Scholar [26] T. Tao, On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), 1-48. doi: 10.4310/DPDE.2004.v1.n1.a1. Google Scholar [27] T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106. Google Scholar [28] C. L. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 134, 25pp. doi: 10.1007/s00526-016-1068-6. Google Scholar

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##### 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 and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equations, preprint, arXiv: 1904.05339.Google Scholar [3] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. Google Scholar [4] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅱ. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556. 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, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. Google Scholar [7] J. Colliander and S. Roudenko, Mass concentration window size and Strichartz norm divergence rate for the L2-critical nonlinear Schrödinger equation, J. Hyperbolic Differ. Equ., 4 (2007), 613-627. doi: 10.1142/S0219891607001288. Google Scholar [8] B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867. doi: 10.1090/proc/13678. Google Scholar [9] 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 [10] 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 [11] 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 [12] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136. doi: 10.1007/BF01214768. Google Scholar [13] C. D. Guevara, Global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, 2 (2014), 177-243. Google Scholar [14] J. Holmer and S. Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, (2007), Art. ID abm004, 31pp. Google Scholar [15] 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 [16] 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 [17] J. Krieger, E. Lenzmann and P. Raphaël, On stability of pseudo-conformal blowup for L2-critical Hartree NLS, Ann. Henri Poincaré, 10 (2009), 1159-1205. doi: 10.1007/s00023-009-0010-2. Google Scholar [18] M. K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p = 0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar [19] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105. doi: 10.1002/sapm197757293. Google Scholar [20] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 18 (1983), 349–374. doi: 10.2307/2007032. Google Scholar [21] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. Google Scholar [22] P.-L. Lions, Compactness and topological methods for some nonlinear variational problems of mathematical physics, In Nonlinear Problems: Present and Future, (Los Alamos, N.M., 1981), North-Holland Math. Stud., 61, North-Holland, Amsterdam-New York, (1982), 17–34. Google Scholar [23] 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. Google Scholar [24] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813. doi: 10.1007/s11784-016-0373-1. Google Scholar [25] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517. Google Scholar [26] T. Tao, On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), 1-48. doi: 10.4310/DPDE.2004.v1.n1.a1. Google Scholar [27] T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106. Google Scholar [28] C. L. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 134, 25pp. doi: 10.1007/s00526-016-1068-6. Google Scholar
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