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,, \emph{Kyoto J. Math.}, 53 (2013), 629. doi: 10.1215/21562261-2265914. Google Scholar

[2]

H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, \emph{Amer. J. Math.}, 121 (1999), 131. Google Scholar

[3]

J. E. Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation,, \emph{J. Math. Phys.}, 25 (1984), 3270. doi: 10.1063/1.526074. Google Scholar

[4]

J. Bergh and J. Löfström, Interpolation spaces. An introduction,, Springer-Verlag, (1976). Google Scholar

[5]

T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics,, New York University Courant Institute of Mathematical Sciences, (2003). Google Scholar

[6]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation,, \emph{Comm. Math. Phys.}, 147 (1992), 75. Google Scholar

[7]

F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation,, \emph{J. Funct. Anal.}, 100 (1991), 87. doi: 10.1016/0022-1236(91)90103-C. Google Scholar

[8]

P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations,, \emph{J. Amer. Math. Soc.}, 1 (1988), 413. 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 {\tt arXiv1104:1114.}, (2011). Google Scholar

[10]

T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation,, \emph{Math. Res. Lett.}, 15 (2008), 1233. 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,, \emph{Sci. China Math.}, 54 (2011), 2037. doi: 10.1007/s11425-011-4283-9. Google Scholar

[12]

G. Fibich, Singular solution of the subcritical nonlinear Schrödinger equation,, \emph{Phys. D}, 240 (2011), 1119. doi: 10.1016/j.physd.2011.04.004. Google Scholar

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D. Foschi, Inhomogeneous Strichartz estimates,, \emph{J. Hyperbolic Differ. Equ.}, 2 (2005), 1. doi: 10.1142/S0219891605000361. Google Scholar

[14]

M. Frazier and B. Jawerth, Decomposition of Besov spaces,, \emph{Indiana Univ. Math. J.}, 34 (1985), 777. doi: 10.1512/iumj.1985.34.34041. Google Scholar

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P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées,, in \emph{S\'eminaire sur les \'Equations aux D\'eriv\'ees Partielles, (1997), 1996. Google Scholar

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J. Ginibre, T. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations,, \emph{Ann. Inst. H. Poincar\'e Phys. Th\'eor.}, 60 (1994), 211. Google Scholar

[17]

K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity,, \emph{Funkcial. Ekvac.}, 51 (2008), 135. 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,, \emph{Comm. Math. Phys.}, 282 (2008), 435. doi: 10.1007/s00220-008-0529-y. Google Scholar

[19]

T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations,, in \emph{Spectral and scattering theory and applications}, (1994), 223. 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,, \emph{Invent. Math.}, 166 (2006), 645. 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,, \emph{Comm. Pure Appl. Math.}, 46 (1993), 527. doi: 10.1002/cpa.3160460405. Google Scholar

[22]

S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations,, \emph{J. Differential Equations}, 175 (2001), 353. 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,, \emph{Amer. J. Math.}, 132 (2010), 361. doi: 10.1353/ajm.0.0107. Google Scholar

[24]

Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation,, \emph{J. Math. Anal. Appl.}, 373 (2011), 147. 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 {\tt 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,, \emph{Calc. Var. Partial Differential Equations}, 44 (2012), 1. doi: 10.1007/s00526-011-0424-9. Google Scholar

[27]

K. Nakanishi, Asymptotically-free solutions for the short-range nonlinear Schrödinger equation,, \emph{SIAM J. Math. Anal.}, 32 (2001), 1265. doi: 10.1137/S0036141000369083. Google Scholar

[28]

K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 9 (2002), 45. doi: 10.1007/s00030-002-8118-9. Google Scholar

[29]

P. Sjölin, Regularity of solutions to the Schrödinger equation,, \emph{Duke Math. J.}, 55 (1987), 699. 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,, \emph{Ann. Inst. H. Poincar\'e Phys. Th\'eor.}, 43 (1985), 321. Google Scholar

[33]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, \emph{Funkcial. Ekvac.}, 30 (1987), 115. Google Scholar

[34]

Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 11 (1984), 186. doi: 10.1090/S0273-0979-1984-15263-7. Google Scholar

[35]

L. Vega, Schrödinger equations: pointwise convergence to the initial data,, \emph{Proc. Amer. Math. Soc.}, 102 (1988), 874. doi: 10.2307/2047326. Google Scholar

[36]

M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation,, \emph{Trans. Amer. Math. Soc.}, 359 (2007), 2123. doi: 10.1090/S0002-9947-06-04099-2. Google Scholar

[37]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions,, \emph{Duke Math. J.}, 138 (2007), 281. 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

show all references

References:
[1]

T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations,, \emph{Kyoto J. Math.}, 53 (2013), 629. doi: 10.1215/21562261-2265914. Google Scholar

[2]

H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, \emph{Amer. J. Math.}, 121 (1999), 131. Google Scholar

[3]

J. E. Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation,, \emph{J. Math. Phys.}, 25 (1984), 3270. doi: 10.1063/1.526074. Google Scholar

[4]

J. Bergh and J. Löfström, Interpolation spaces. An introduction,, Springer-Verlag, (1976). Google Scholar

[5]

T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics,, New York University Courant Institute of Mathematical Sciences, (2003). Google Scholar

[6]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation,, \emph{Comm. Math. Phys.}, 147 (1992), 75. Google Scholar

[7]

F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation,, \emph{J. Funct. Anal.}, 100 (1991), 87. doi: 10.1016/0022-1236(91)90103-C. Google Scholar

[8]

P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations,, \emph{J. Amer. Math. Soc.}, 1 (1988), 413. 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 {\tt arXiv1104:1114.}, (2011). Google Scholar

[10]

T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation,, \emph{Math. Res. Lett.}, 15 (2008), 1233. 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,, \emph{Sci. China Math.}, 54 (2011), 2037. doi: 10.1007/s11425-011-4283-9. Google Scholar

[12]

G. Fibich, Singular solution of the subcritical nonlinear Schrödinger equation,, \emph{Phys. D}, 240 (2011), 1119. doi: 10.1016/j.physd.2011.04.004. Google Scholar

[13]

D. Foschi, Inhomogeneous Strichartz estimates,, \emph{J. Hyperbolic Differ. Equ.}, 2 (2005), 1. doi: 10.1142/S0219891605000361. Google Scholar

[14]

M. Frazier and B. Jawerth, Decomposition of Besov spaces,, \emph{Indiana Univ. Math. J.}, 34 (1985), 777. doi: 10.1512/iumj.1985.34.34041. Google Scholar

[15]

P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées,, in \emph{S\'eminaire sur les \'Equations aux D\'eriv\'ees Partielles, (1997), 1996. Google Scholar

[16]

J. Ginibre, T. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations,, \emph{Ann. Inst. H. Poincar\'e Phys. Th\'eor.}, 60 (1994), 211. Google Scholar

[17]

K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity,, \emph{Funkcial. Ekvac.}, 51 (2008), 135. 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,, \emph{Comm. Math. Phys.}, 282 (2008), 435. doi: 10.1007/s00220-008-0529-y. Google Scholar

[19]

T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations,, in \emph{Spectral and scattering theory and applications}, (1994), 223. 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,, \emph{Invent. Math.}, 166 (2006), 645. 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,, \emph{Comm. Pure Appl. Math.}, 46 (1993), 527. doi: 10.1002/cpa.3160460405. Google Scholar

[22]

S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations,, \emph{J. Differential Equations}, 175 (2001), 353. 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,, \emph{Amer. J. Math.}, 132 (2010), 361. doi: 10.1353/ajm.0.0107. Google Scholar

[24]

Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation,, \emph{J. Math. Anal. Appl.}, 373 (2011), 147. 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 {\tt 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,, \emph{Calc. Var. Partial Differential Equations}, 44 (2012), 1. doi: 10.1007/s00526-011-0424-9. Google Scholar

[27]

K. Nakanishi, Asymptotically-free solutions for the short-range nonlinear Schrödinger equation,, \emph{SIAM J. Math. Anal.}, 32 (2001), 1265. doi: 10.1137/S0036141000369083. Google Scholar

[28]

K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 9 (2002), 45. doi: 10.1007/s00030-002-8118-9. Google Scholar

[29]

P. Sjölin, Regularity of solutions to the Schrödinger equation,, \emph{Duke Math. J.}, 55 (1987), 699. 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,, \emph{Ann. Inst. H. Poincar\'e Phys. Th\'eor.}, 43 (1985), 321. Google Scholar

[33]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, \emph{Funkcial. Ekvac.}, 30 (1987), 115. Google Scholar

[34]

Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 11 (1984), 186. doi: 10.1090/S0273-0979-1984-15263-7. Google Scholar

[35]

L. Vega, Schrödinger equations: pointwise convergence to the initial data,, \emph{Proc. Amer. Math. Soc.}, 102 (1988), 874. doi: 10.2307/2047326. Google Scholar

[36]

M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation,, \emph{Trans. Amer. Math. Soc.}, 359 (2007), 2123. doi: 10.1090/S0002-9947-06-04099-2. Google Scholar

[37]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions,, \emph{Duke Math. J.}, 138 (2007), 281. 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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