# American Institute of Mathematical Sciences

July  2015, 14(4): 1481-1531. doi: 10.3934/cpaa.2015.14.1481

## A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation

 1 Laboratory of Mathematics, Institute of Engineering, Hiroshima university, Higashihiroshima Hirhosima, 739-8527, Japan

Received  September 2013 Revised  May 2014 Published  April 2015

This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schrödinger equation of power type with data in a critical homogeneous weighted $L^2$ space. We give a sharp sufficient condition for scattering by proving existence of a threshold solution which does not scatter at least for one time direction and of which initial data attains minimum value of a norm of the weighted $L^2$ space in all initial value of non-scattering solution. Unlike in the mass-critical or -supercritical case, ground state is not a threshold. This is an extension of previous author's result to the case where the exponent of nonlinearity is below so-called Strauss number. A main new ingredient is a stability estimate in a Lorenz-modified-Bezov type spacetime norm.
Citation: 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
##### 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.  Google Scholar [2] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.  Google Scholar [3] J. E. Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273. doi: 10.1063/1.526074.  Google Scholar [4] J. Bergh and J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.  Google Scholar [5] T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003. Google Scholar [6] T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.  Google Scholar [7] F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.  Google Scholar [8] P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439. doi: 10.2307/1990923.  Google Scholar [9] B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, archived as arXiv1104:1114., 2011. Google Scholar [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.  Google Scholar [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.  Google Scholar [12] G. Fibich, Singular solution of the subcritical nonlinear Schrödinger equation, Phys. D, 240 (2011), 1119-1122. doi: 10.1016/j.physd.2011.04.004.  Google Scholar [13] D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361.  Google Scholar [14] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34 (1985), 777-799. doi: 10.1512/iumj.1985.34.34041.  Google Scholar [15] P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées, in Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, École Polytech., Palaiseau, 1997, Exp. No. IV, 11.  Google Scholar [16] J. Ginibre, T. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 211-239.  Google Scholar [17] K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147. doi: 10.1619/fesi.51.135.  Google Scholar [18] 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.  Google Scholar [19] T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, vol. 23 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 1994, 223-238.  Google Scholar [20] 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 [21] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar [22] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.  Google Scholar [23] R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107.  Google Scholar [24] Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, J. Math. Anal. Appl., 373 (2011), 147-160. doi: 10.1016/j.jmaa.2010.06.019.  Google Scholar [25] S. Masaki, On minimal non-scattering solution to focusing mass-subcritical nonlinear Schrödinger equation, archived as arXiv:1301.1742., 2013. Google Scholar [26] K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations, 44 (2012), 1-45. doi: 10.1007/s00526-011-0424-9.  Google Scholar [27] K. Nakanishi, Asymptotically-free solutions for the short-range nonlinear Schrödinger equation, SIAM J. Math. Anal., 32 (2001), 1265-1271 (electronic). doi: 10.1137/S0036141000369083.  Google Scholar [28] K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 45-68. doi: 10.1007/s00030-002-8118-9.  Google Scholar [29] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J., 55 (1987), 699-715. doi: 10.1215/S0012-7094-87-05535-9.  Google Scholar [30] W. A. Strauss, Nonlinear scattering theory,, \emph{Scattering Theory in Mathematical Physics, (): 53.   Google Scholar [31] T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions,, \emph{Electron. J. Differential Equations}, ().   Google Scholar [32] Y. Tsutsumi, Scattering problem for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 43 (1985), 321-347.  Google Scholar [33] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.  Google Scholar [34] Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc. (N.S.), 11 (1984), 186-188. doi: 10.1090/S0273-0979-1984-15263-7.  Google Scholar [35] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878. doi: 10.2307/2047326.  Google Scholar [36] M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136 (electronic). doi: 10.1090/S0002-9947-06-04099-2.  Google Scholar [37] 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 [38] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, \emph{Comm. Math. Phys.}, 87 (): 567.   Google Scholar

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##### 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.  Google Scholar [2] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.  Google Scholar [3] J. E. Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273. doi: 10.1063/1.526074.  Google Scholar [4] J. Bergh and J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.  Google Scholar [5] T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003. Google Scholar [6] T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.  Google Scholar [7] F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.  Google Scholar [8] P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439. doi: 10.2307/1990923.  Google Scholar [9] B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, archived as arXiv1104:1114., 2011. Google Scholar [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.  Google Scholar [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.  Google Scholar [12] G. Fibich, Singular solution of the subcritical nonlinear Schrödinger equation, Phys. D, 240 (2011), 1119-1122. doi: 10.1016/j.physd.2011.04.004.  Google Scholar [13] D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361.  Google Scholar [14] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34 (1985), 777-799. doi: 10.1512/iumj.1985.34.34041.  Google Scholar [15] P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées, in Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, École Polytech., Palaiseau, 1997, Exp. No. IV, 11.  Google Scholar [16] J. Ginibre, T. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 211-239.  Google Scholar [17] K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147. doi: 10.1619/fesi.51.135.  Google Scholar [18] 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.  Google Scholar [19] T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, vol. 23 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 1994, 223-238.  Google Scholar [20] 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 [21] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar [22] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.  Google Scholar [23] R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107.  Google Scholar [24] Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, J. Math. Anal. Appl., 373 (2011), 147-160. doi: 10.1016/j.jmaa.2010.06.019.  Google Scholar [25] S. Masaki, On minimal non-scattering solution to focusing mass-subcritical nonlinear Schrödinger equation, archived as arXiv:1301.1742., 2013. Google Scholar [26] K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations, 44 (2012), 1-45. doi: 10.1007/s00526-011-0424-9.  Google Scholar [27] K. Nakanishi, Asymptotically-free solutions for the short-range nonlinear Schrödinger equation, SIAM J. Math. Anal., 32 (2001), 1265-1271 (electronic). doi: 10.1137/S0036141000369083.  Google Scholar [28] K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 45-68. doi: 10.1007/s00030-002-8118-9.  Google Scholar [29] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J., 55 (1987), 699-715. doi: 10.1215/S0012-7094-87-05535-9.  Google Scholar [30] W. A. Strauss, Nonlinear scattering theory,, \emph{Scattering Theory in Mathematical Physics, (): 53.   Google Scholar [31] T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions,, \emph{Electron. J. Differential Equations}, ().   Google Scholar [32] Y. Tsutsumi, Scattering problem for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 43 (1985), 321-347.  Google Scholar [33] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.  Google Scholar [34] Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc. (N.S.), 11 (1984), 186-188. doi: 10.1090/S0273-0979-1984-15263-7.  Google Scholar [35] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878. doi: 10.2307/2047326.  Google Scholar [36] M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136 (electronic). doi: 10.1090/S0002-9947-06-04099-2.  Google Scholar [37] 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 [38] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, \emph{Comm. Math. Phys.}, 87 (): 567.   Google Scholar
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