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Scattering of radial data in the focusing NLS and generalized Hartree equations
Department of Mathematics & Statistics, Florida International University, Miami, FL 33199, USA |
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 [
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 [
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 and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equations, preprint, arXiv: 1904.05339. |
[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. |
[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. |
[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, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
P.-L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[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. |
[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. |
[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. |
[25] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[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. |
[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. |
[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. |
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 and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equations, preprint, arXiv: 1904.05339. |
[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. |
[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. |
[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, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
P.-L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[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. |
[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. |
[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. |
[25] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[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. |
[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. |
[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. |
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