A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system

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March 2021, 41(3): 1415-1447. doi: 10.3934/dcds.2020323

A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system

Masaru Hamano ^1,, and Satoshi Masaki ^2,

1.	Department of Mathematics, Graduate School of Science and Engineering, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama-shi, Saitama, 338-8570, Japan
2.	Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan

^* Corresponding author: Masaru Hamano

The first author is supported by JSPS KAKENHI Grant Number JP19J13300

Received December 2019 Revised June 2020 Published March 2021 Early access September 2020

Fund Project: The second author is supported by JSPS KAKENHI Grant Numbers JP17K14219, JP17H02854, JP17H02851, and JP18KK0386

Abstract
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In this paper, we consider the quadratic nonlinear Schrödinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criterion is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of the zero function and a linear Schrödinger flow. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.

Keywords: Nonlinear Schrödinger system, mass-subcritical, scattering.

Mathematics Subject Classification: Primary: 35Q55.

Citation: Masaru Hamano, Satoshi Masaki. A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1415-1447. doi: 10.3934/dcds.2020323

References:

[1]	T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529.
[2]	M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, preprint, arXiv: math/0311048.
[3]	M. Colin, Th. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2211-2226. doi: 10.1016/j.anihpc.2009.01.011.
[4]	V. D. Dinh, Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction, Nonlinear Anal., 190 (2020), 111589, 39 pp. doi: 10.1016/j.na.2019.111589.
[5]	B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618. doi: 10.1016/j.aim.2015.04.030.
[6]	J. Ginibre, T. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincar Phys. Théor., 60 (1994), 211-239.
[7]	J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188. doi: 10.1007/BF02099195.
[8]	M. Hamano, Global dynamics below the ground state for the quadratic Schrödinger system in 5d, preprint, arXiv: 1805.12245.
[9]	N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations, Differ. Equ. Appl., 3 (2011), 415-426. doi: 10.7153/dea-03-26.
[10]	N. Hayashi, T. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690. doi: 10.1016/j.anihpc.2012.10.007.
[11]	R. A. Hunt, On $L(p, q)$ spaces, Enseignement Math. (2), 12 (1966), 249–276.
[12]	T. Inui, N. Kishimoto and K. Nishimura, Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition, Discrete Contin. Dyn. Syst., 39 (2019), 6299-6353. doi: 10.3934/dcds.2019275.
[13]	T. Kato, An $L^{q, r}$-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications, Adv. Stud. Pure Math., 23, Math. Soc. Japan, Tokyo, (1994), 223–238. doi: 10.2969/aspm/02310223.
[14]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[15]	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.
[16]	R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 38, 33 pp. doi: 10.1007/s00030-017-0463-9.
[17]	R. Killip, S. Masaki, J. Murphy and M. Visan, The radial mass-subcritical NLS in negative order Sobolev spaces, Discrete Contin. Dyn. Syst., 39 (2019), 553-583. doi: 10.3934/dcds.2019023.
[18]	R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203-1258. doi: 10.4171/JEMS/180.
[19]	R. Killip, M. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266. doi: 10.2140/apde.2008.1.229.
[20]	S. Masaki, A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 14 (2015), 1481-1531. doi: 10.3934/cpaa.2015.14.1481.
[21]	S. Masaki, On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation, Comm. Partial Differential Equations, 42 (2017), 626-653. doi: 10.1080/03605302.2017.1286672.
[22]	S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, preprint, arXiv: 1605.09234.
[23]	S. Masaki and J.-I. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 283-326. doi: 10.1016/j.anihpc.2017.04.003.
[24]	K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrodinger equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 45-68. doi: 10.1007/s00030-002-8118-9.
[25]	R. O'Neil, Convolution operators and L(p, q) spaces, Duke Math. J., 30 (1963), 129-142. doi: 10.1215/S0012-7094-63-03015-1.
[26]	R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714. doi: 10.1215/S0012-7094-77-04430-1.

show all references

References:

[1]	T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529.
[2]	M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, preprint, arXiv: math/0311048.
[3]	M. Colin, Th. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2211-2226. doi: 10.1016/j.anihpc.2009.01.011.
[4]	V. D. Dinh, Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction, Nonlinear Anal., 190 (2020), 111589, 39 pp. doi: 10.1016/j.na.2019.111589.
[5]	B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618. doi: 10.1016/j.aim.2015.04.030.
[6]	J. Ginibre, T. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincar Phys. Théor., 60 (1994), 211-239.
[7]	J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188. doi: 10.1007/BF02099195.
[8]	M. Hamano, Global dynamics below the ground state for the quadratic Schrödinger system in 5d, preprint, arXiv: 1805.12245.
[9]	N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations, Differ. Equ. Appl., 3 (2011), 415-426. doi: 10.7153/dea-03-26.
[10]	N. Hayashi, T. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690. doi: 10.1016/j.anihpc.2012.10.007.
[11]	R. A. Hunt, On $L(p, q)$ spaces, Enseignement Math. (2), 12 (1966), 249–276.
[12]	T. Inui, N. Kishimoto and K. Nishimura, Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition, Discrete Contin. Dyn. Syst., 39 (2019), 6299-6353. doi: 10.3934/dcds.2019275.
[13]	T. Kato, An $L^{q, r}$-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications, Adv. Stud. Pure Math., 23, Math. Soc. Japan, Tokyo, (1994), 223–238. doi: 10.2969/aspm/02310223.
[14]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[15]	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.
[16]	R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 38, 33 pp. doi: 10.1007/s00030-017-0463-9.
[17]	R. Killip, S. Masaki, J. Murphy and M. Visan, The radial mass-subcritical NLS in negative order Sobolev spaces, Discrete Contin. Dyn. Syst., 39 (2019), 553-583. doi: 10.3934/dcds.2019023.
[18]	R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203-1258. doi: 10.4171/JEMS/180.
[19]	R. Killip, M. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266. doi: 10.2140/apde.2008.1.229.
[20]	S. Masaki, A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 14 (2015), 1481-1531. doi: 10.3934/cpaa.2015.14.1481.
[21]	S. Masaki, On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation, Comm. Partial Differential Equations, 42 (2017), 626-653. doi: 10.1080/03605302.2017.1286672.
[22]	S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, preprint, arXiv: 1605.09234.
[23]	S. Masaki and J.-I. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 283-326. doi: 10.1016/j.anihpc.2017.04.003.
[24]	K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrodinger equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 45-68. doi: 10.1007/s00030-002-8118-9.
[25]	R. O'Neil, Convolution operators and L(p, q) spaces, Duke Math. J., 30 (1963), 129-142. doi: 10.1215/S0012-7094-63-03015-1.
[26]	R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714. doi: 10.1215/S0012-7094-77-04430-1.

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