# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020323

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

 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  September 2020

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

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.

Citation: Masaru Hamano, Satoshi Masaki. A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar [2] M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, preprint, arXiv: math/0311048. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [8] M. Hamano, Global dynamics below the ground state for the quadratic Schrödinger system in 5d, preprint, arXiv: 1805.12245. Google Scholar [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.  Google Scholar [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.  Google Scholar [11] R. A. Hunt, On $L(p, q)$ spaces, Enseignement Math. (2), 12 (1966), 249–276.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, preprint, arXiv: 1605.09234. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [2] M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, preprint, arXiv: math/0311048. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [8] M. Hamano, Global dynamics below the ground state for the quadratic Schrödinger system in 5d, preprint, arXiv: 1805.12245. Google Scholar [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.  Google Scholar [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.  Google Scholar [11] R. A. Hunt, On $L(p, q)$ spaces, Enseignement Math. (2), 12 (1966), 249–276.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, preprint, arXiv: 1605.09234. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481 [2] Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073 [3] Rowan Killip, Satoshi Masaki, Jason Murphy, Monica Visan. The radial mass-subcritical NLS in negative order Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 553-583. doi: 10.3934/dcds.2019023 [4] Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003 [5] Thierry Cazenave, Zheng Han. Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4801-4819. doi: 10.3934/dcds.2020202 [6] Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563 [7] Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129 [8] Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803 [9] Van Duong Dinh. A unified approach for energy scattering for focusing nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6441-6471. doi: 10.3934/dcds.2020286 [10] Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565 [11] Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117 [12] Benedetta Noris, Hugo Tavares, Gianmaria Verzini. Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6085-6112. doi: 10.3934/dcds.2015.35.6085 [13] Myeongju Chae, Sunggeum Hong, Sanghyuk Lee. Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 909-928. doi: 10.3934/dcds.2011.29.909 [14] Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034 [15] Jianjun Yuan. Global existence and scattering of equivariant defocusing Chern-Simons-Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5541-5570. doi: 10.3934/dcds.2020237 [16] Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183 [17] Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831 [18] Van Duong Dinh, Sahbi Keraani. The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020407 [19] J. Colliander, Justin Holmer, Monica Visan, Xiaoyi Zhang. Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $R$. Communications on Pure & Applied Analysis, 2008, 7 (3) : 467-489. doi: 10.3934/cpaa.2008.7.467 [20] Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091

2019 Impact Factor: 1.338

Article outline