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

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.
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. ColinT. 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. CollianderM. KeelG. StaffilaniH. 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. DuyckaertsJ. 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. HayashiC. 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. HayashiT. 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. TaoM. 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. TaoM. 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. TaoM. 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. ColinT. 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. CollianderM. KeelG. StaffilaniH. 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. DuyckaertsJ. 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. HayashiC. 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. HayashiT. 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. TaoM. 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. TaoM. 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. TaoM. 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

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