American Institute of Mathematical Sciences

Advanced
November  2019, 39(11): 6299-6353. doi: 10.3934/dcds.2019275

Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition

Takahisa Inui 1, , Nobu Kishimoto 2, and  Kuranosuke Nishimura 3,,

1. 

Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
2. 

Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan
3. 

Department of Mathematics, Graduate School of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

* Corresponding author: Kuranosuke Nishimura

Received  November 2018 Published  August 2019

Fund Project: The first author is partially supported by JSPS Grant-in-Aid for Early-Career Scientists JP18K13444. The second author is supported in part by Grant-in-Aid for Young Scientists (B) JP24740086 and JP16K17626

We consider a mass-critical system of nonlinear Schrödinger equations
$ \left\{ \begin{split} i\partial_t u + \;\; \Delta u & = \bar{u}v,\\ i\partial_t v +\kappa \Delta v & = u^2, \end{split} \right. \qquad (t,x)\in \mathbb{R}\times \mathbb{R}^4, $
where
$ (u,v) $
 is a
$ \mathbb{C}^2 $
-valued unknown function and
$ \kappa >0 $
 is a constant. If
$ \kappa = 1/2 $
, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition
$ M(u,v)<M(\phi ,\psi) $
, where
$ M(u,v) $
 denotes the mass and
$ (\phi ,\psi) $
 is a ground state. In the mass-resonance case, we prove scattering by the argument of Dodson [5]. Scattering is also obtained without mass-resonance condition under the restriction that
$ (u,v) $
 is radially symmetric.
Keywords: Quadratic NLS system, mass critical, scattering, mass-resonance, concentration compactness.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35P25.
Citation: Takahisa Inui, Nobu Kishimoto, Kuranosuke Nishimura. Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6299-6353. doi: 10.3934/dcds.2019275
References:
[1]

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

[2]

M. Colin, T. 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

[3]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math.(2), 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. Google Scholar

[4]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2$-critical nonlinear Schrödinger equation when $d\geq3$, J. Amer. Math. Soc., 25 (2012), 429-463. doi: 10.1090/S0894-0347-2011-00727-3. 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]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2$ critical, nonlinear Schrödinger equation when $d = 1$, Amer. J. Math., 138 (2016), 531-569. doi: 10.1353/ajm.2016.0016. Google Scholar

[7]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schrödinger equation when $d = 2$, Duke Math. J., 165 (2016), 3435-3516. doi: 10.1215/00127094-3673888. Google Scholar

[8]

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

[9]

M. Hamano, Global dynamics below the ground state for the quadratic schödinger system in $5d$, preprint, arXiv: 1805.12245, 2018.Google Scholar

[10]

N. Hayashi, C. H. 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

[11]

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

[12]

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

[13]

R. Killip and M. Vișan, Nonlinear schrödinger equations at critical regularity, Evolution equations, Clay Math. Proc., Amer. Math. Soc., Providence, RI, 17 (2013), 325-437. Google Scholar

[14]

H. Koch, D. Tataru and M. Vișan, Dispersive Equations and Nonlinear Waves, Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps. Oberwolfach Seminars, 45. Birkhäuser/Springer, Basel, 2014. Google Scholar

[15]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbf{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar

[16]

F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, (1998), 399-425. Google Scholar

[17]

T. Ozawa and H. Sunagawa, Small data blow-up for a system of nonlinear Schrödinger equations, J. Math. Anal. Appl., 399 (2013), 147-155. doi: 10.1016/j.jmaa.2012.10.003. Google Scholar

[18]

T. Tao, M. Visan and X. Y. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., 140 (2007), 165-202. doi: 10.1215/S0012-7094-07-14015-8. Google Scholar

[19]

T. Tao, M. Visan and X. Y. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805. Google Scholar

[20]

T. Tao, M. Visan and X. Y. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math., 20 (2008), 881-919. doi: 10.1515/FORUM.2008.042. Google Scholar

[21]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374. doi: 10.1215/S0012-7094-07-13825-0. Google Scholar

show all references

References:
[1]

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

[2]

M. Colin, T. 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

[3]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math.(2), 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. Google Scholar

[4]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2$-critical nonlinear Schrödinger equation when $d\geq3$, J. Amer. Math. Soc., 25 (2012), 429-463. doi: 10.1090/S0894-0347-2011-00727-3. 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]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2$ critical, nonlinear Schrödinger equation when $d = 1$, Amer. J. Math., 138 (2016), 531-569. doi: 10.1353/ajm.2016.0016. Google Scholar

[7]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schrödinger equation when $d = 2$, Duke Math. J., 165 (2016), 3435-3516. doi: 10.1215/00127094-3673888. Google Scholar

[8]

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

[9]

M. Hamano, Global dynamics below the ground state for the quadratic schödinger system in $5d$, preprint, arXiv: 1805.12245, 2018.Google Scholar

[10]

N. Hayashi, C. H. 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

[11]

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

[12]

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

[13]

R. Killip and M. Vișan, Nonlinear schrödinger equations at critical regularity, Evolution equations, Clay Math. Proc., Amer. Math. Soc., Providence, RI, 17 (2013), 325-437. Google Scholar

[14]

H. Koch, D. Tataru and M. Vișan, Dispersive Equations and Nonlinear Waves, Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps. Oberwolfach Seminars, 45. Birkhäuser/Springer, Basel, 2014. Google Scholar

[15]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbf{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar

[16]

F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, (1998), 399-425. Google Scholar

[17]

T. Ozawa and H. Sunagawa, Small data blow-up for a system of nonlinear Schrödinger equations, J. Math. Anal. Appl., 399 (2013), 147-155. doi: 10.1016/j.jmaa.2012.10.003. Google Scholar

[18]

T. Tao, M. Visan and X. Y. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., 140 (2007), 165-202. doi: 10.1215/S0012-7094-07-14015-8. Google Scholar

[19]

T. Tao, M. Visan and X. Y. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805. Google Scholar

[20]

T. Tao, M. Visan and X. Y. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math., 20 (2008), 881-919. doi: 10.1515/FORUM.2008.042. Google Scholar

[21]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374. doi: 10.1215/S0012-7094-07-13825-0. Google Scholar

[1]

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
[2]

Yanfang Gao, Zhiyong Wang. Minimal mass non-scattering solutions of the focusing L2-critical Hartree equations with radial data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1979-2007. doi: 10.3934/dcds.2017084
[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]

Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066
[5]

Casey Jao. Energy-critical NLS with potentials of quadratic growth. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 563-587. doi: 10.3934/dcds.2018025
[6]

Rowan Killip, Soonsik Kwon, Shuanglin Shao, Monica Visan. On the mass-critical generalized KdV equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 191-221. doi: 10.3934/dcds.2012.32.191
[7]

Tong Li, Hailiang Liu. Critical thresholds in a relaxation system with resonance of characteristic speeds. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 511-521. doi: 10.3934/dcds.2009.24.511
[8]

Ruihong Ji, Yongfu Wang. Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1117-1133. doi: 10.3934/dcds.2019047
[9]

M. Pellicer, J. Solà-Morales. Spectral analysis and limit behaviours in a spring-mass system. Communications on Pure & Applied Analysis, 2008, 7 (3) : 563-577. doi: 10.3934/cpaa.2008.7.563
[10]

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
[11]

Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106
[12]

David Kinderlehrer, Adrian Tudorascu. Transport via mass transportation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 311-338. doi: 10.3934/dcdsb.2006.6.311
[13]

Dennis L. Chao, Dobromir T. Dimitrov. Seasonality and the effectiveness of mass vaccination. Mathematical Biosciences & Engineering, 2016, 13 (2) : 249-259. doi: 10.3934/mbe.2015001
[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]

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
[16]

Elio E. Espejo, Masaki Kurokiba, Takashi Suzuki. Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2627-2644. doi: 10.3934/cpaa.2013.12.2627
[17]

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
[18]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[19]

Belinda A. Batten, Hesam Shoori, John R. Singler, Madhuka H. Weerasinghe. Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 83-107. doi: 10.3934/dcdsb.2018162
[20]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (32)
  • HTML views (53)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences