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April  2017, 37(4): 2077-2102. doi: 10.3934/dcds.2017089

Almost global existence for cubic nonlinear Schrödinger equations in one space dimension

Jason Murphy 1,, and  Fabio Pusateri 2,

1. 

Department of Mathematics, University of California, 970 Evans Hall, Berkeley, CA 94720-3840, USA
2. 

Department of Mathematics, Princeton University, Fine Hall, 304 Washington Rd, Princeton, NJ 08544, USA

* Corresponding author: Jason Murphy

Received  May 2016 Revised  November 2016 Published  December 2016

We consider non-gauge-invariant cubic nonlinear Schrödinger equations in one space dimension.We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^∞$ decay up to time $\exp(C\varepsilon^{-2})$. We also exhibit norm growth beyond this time for a specific choice of nonlinearity.

Keywords: Cubic NLS, almost global existence, method of space-time resonances.
Mathematics Subject Classification: Primary:35Q55.
Citation: Jason Murphy, Fabio Pusateri. Almost global existence for cubic nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2077-2102. doi: 10.3934/dcds.2017089
References:
[1]

J. Barab, Nonexistence of asymptotically free solutions of a nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.  doi: 10.1063/1.526074.  Google Scholar

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp. doi: 10.1090/cln/010.  Google Scholar

[3]

R. Coifman and Y. Meyer, Ondelettes et Opérateurs. Ⅲ. Opérateurs Multilinéaires, Actualités Mathématiques. Hermann, Paris, 1991.  Google Scholar

[4]

P. Deift and X. Zhou, Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Dedicated to the memory of Jürgen K. Moser, Comm. Pure Appl. Math., 56 (2003), 1029-1077.  doi: 10.1002/cpa.3034.  Google Scholar

[5]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN, (2009), 414-432.  doi: 10.1093/imrn/rnn135.  Google Scholar

[6]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for 2D quadratic Schrödinger equations, J. Math. Pures Appl.(9), 97 (2012), 505-543.  doi: 10.1016/j.matpur.2011.09.008.  Google Scholar

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N. Hayashi and P. Naumkin, Asymptotics for large time of solutions to nonlinear Schrödinger and Hartree equations, Amer. J. Math., 20 (1998), 369-389.  doi: 10.1353/ajm.1998.0011.  Google Scholar

[8]

N. Hayashi and P. Naumkin, Asymptotics of small solutions to nonlinear Schrödinger equations with cubic nonlinearities, Int. J. Pure Appl. Math., 3 (2002), 255-273.   Google Scholar

[9]

N. Hayashi and P. Naumkin, Large time behavior for the cubic nonlinear Schrödinger equation, Canad. J. Math., 54 (2002), 1065-1085.  doi: 10.4153/CJM-2002-039-3.  Google Scholar

[10]

N. Hayashi and P. Naumkin, On the asymptotics for cubic nonlinear Schrödinger equations, Complex Var. Theory Appl., 49 (2004), 339-373.  doi: 10.1080/02781070410001710353.  Google Scholar

[11]

N. Hayashi and P. Naumkin, Nongauge invariant cubic nonlinear Schrödinger equations, Pac. J. Appl. Math., 1 (2008), 1-16.   Google Scholar

[12]

N. Hayashi and P. Naumkin, Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces, Differential Integral Equations, 24 (2011), 801-828.   Google Scholar

[13]

N. Hayashi and P. Naumkin, Logarithmic time decay for the cubic nonlinear Schrödinger equations, Int Math Res Notices, 2015 (2015), 5604-5643.  doi: 10.1093/imrn/rnu102.  Google Scholar

[14]

M. Ifrim and D. Tataru, Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension, Nonlinearity, 28 (2015), 2661-2675.  doi: 10.1088/0951-7715/28/8/2661.  Google Scholar

[15]

F. John, Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data, Comm. Pure Appl. Math., 40 (1987), 79-109.  doi: 10.1002/cpa.3160400104.  Google Scholar

[16]

F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455.  doi: 10.1002/cpa.3160370403.  Google Scholar

[17]

J. Kato and F. Pusateri, A new proof of long-range scattering for critical nonlinear Schrödinger equations, Differential Integral Equations, 24 (2011), 923-940.   Google Scholar

[18]

H. Lindblad and A. Soffer, Scattering and small data completeness for the critical nonlinear Schrödinger equation, Nonlinearity, 19 (2006), 345-353.  doi: 10.1088/0951-7715/19/2/006.  Google Scholar

[19]

C. Muscalu, J. Pipher, T. Tao and C. Thiele, A Short Proof of the Coifman-Meyer Multilinear Theorem, http://www.math.brown.edu/~jpipher/trilogy1.pdf Google Scholar

[20]

P. Naumkin, Cubic derivative nonlinear Schrödinger equations, SUT J. Math., 36 (2000), 9-42.   Google Scholar

[21]

F. Pusateri and J. Shatah, Space-time resonances and the null condition for first-order systems of wave equations, Comm. Pure Appl. Math., 66 (2013), 1495-1540.  doi: 10.1002/cpa.21461.  Google Scholar

[22]

Y. Sagawa and H. Sunagawa, The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension, Discrete Contin. Dyn. Syst., 36 (2016), 5743-5761.  doi: 10.3934/dcds.2016052.  Google Scholar

[23]

A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations, 31 (2006), 1407-1423.  doi: 10.1080/03605300600910316.  Google Scholar

[24]

H. Sunagawa, Lower bounds of the lifespan of small data solutions to the nonlinear Schrödinger equations, Osaka J. Math., 43 (2006), 771-789.   Google Scholar

[25]

Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc., 11 (1984), 186-188.  doi: 10.1090/S0273-0979-1984-15263-7.  Google Scholar

show all references

References:
[1]

J. Barab, Nonexistence of asymptotically free solutions of a nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.  doi: 10.1063/1.526074.  Google Scholar

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp. doi: 10.1090/cln/010.  Google Scholar

[3]

R. Coifman and Y. Meyer, Ondelettes et Opérateurs. Ⅲ. Opérateurs Multilinéaires, Actualités Mathématiques. Hermann, Paris, 1991.  Google Scholar

[4]

P. Deift and X. Zhou, Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Dedicated to the memory of Jürgen K. Moser, Comm. Pure Appl. Math., 56 (2003), 1029-1077.  doi: 10.1002/cpa.3034.  Google Scholar

[5]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN, (2009), 414-432.  doi: 10.1093/imrn/rnn135.  Google Scholar

[6]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for 2D quadratic Schrödinger equations, J. Math. Pures Appl.(9), 97 (2012), 505-543.  doi: 10.1016/j.matpur.2011.09.008.  Google Scholar

[7]

N. Hayashi and P. Naumkin, Asymptotics for large time of solutions to nonlinear Schrödinger and Hartree equations, Amer. J. Math., 20 (1998), 369-389.  doi: 10.1353/ajm.1998.0011.  Google Scholar

[8]

N. Hayashi and P. Naumkin, Asymptotics of small solutions to nonlinear Schrödinger equations with cubic nonlinearities, Int. J. Pure Appl. Math., 3 (2002), 255-273.   Google Scholar

[9]

N. Hayashi and P. Naumkin, Large time behavior for the cubic nonlinear Schrödinger equation, Canad. J. Math., 54 (2002), 1065-1085.  doi: 10.4153/CJM-2002-039-3.  Google Scholar

[10]

N. Hayashi and P. Naumkin, On the asymptotics for cubic nonlinear Schrödinger equations, Complex Var. Theory Appl., 49 (2004), 339-373.  doi: 10.1080/02781070410001710353.  Google Scholar

[11]

N. Hayashi and P. Naumkin, Nongauge invariant cubic nonlinear Schrödinger equations, Pac. J. Appl. Math., 1 (2008), 1-16.   Google Scholar

[12]

N. Hayashi and P. Naumkin, Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces, Differential Integral Equations, 24 (2011), 801-828.   Google Scholar

[13]

N. Hayashi and P. Naumkin, Logarithmic time decay for the cubic nonlinear Schrödinger equations, Int Math Res Notices, 2015 (2015), 5604-5643.  doi: 10.1093/imrn/rnu102.  Google Scholar

[14]

M. Ifrim and D. Tataru, Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension, Nonlinearity, 28 (2015), 2661-2675.  doi: 10.1088/0951-7715/28/8/2661.  Google Scholar

[15]

F. John, Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data, Comm. Pure Appl. Math., 40 (1987), 79-109.  doi: 10.1002/cpa.3160400104.  Google Scholar

[16]

F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455.  doi: 10.1002/cpa.3160370403.  Google Scholar

[17]

J. Kato and F. Pusateri, A new proof of long-range scattering for critical nonlinear Schrödinger equations, Differential Integral Equations, 24 (2011), 923-940.   Google Scholar

[18]

H. Lindblad and A. Soffer, Scattering and small data completeness for the critical nonlinear Schrödinger equation, Nonlinearity, 19 (2006), 345-353.  doi: 10.1088/0951-7715/19/2/006.  Google Scholar

[19]

C. Muscalu, J. Pipher, T. Tao and C. Thiele, A Short Proof of the Coifman-Meyer Multilinear Theorem, http://www.math.brown.edu/~jpipher/trilogy1.pdf Google Scholar

[20]

P. Naumkin, Cubic derivative nonlinear Schrödinger equations, SUT J. Math., 36 (2000), 9-42.   Google Scholar

[21]

F. Pusateri and J. Shatah, Space-time resonances and the null condition for first-order systems of wave equations, Comm. Pure Appl. Math., 66 (2013), 1495-1540.  doi: 10.1002/cpa.21461.  Google Scholar

[22]

Y. Sagawa and H. Sunagawa, The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension, Discrete Contin. Dyn. Syst., 36 (2016), 5743-5761.  doi: 10.3934/dcds.2016052.  Google Scholar

[23]

A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations, 31 (2006), 1407-1423.  doi: 10.1080/03605300600910316.  Google Scholar

[24]

H. Sunagawa, Lower bounds of the lifespan of small data solutions to the nonlinear Schrödinger equations, Osaka J. Math., 43 (2006), 771-789.   Google Scholar

[25]

Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc., 11 (1984), 186-188.  doi: 10.1090/S0273-0979-1984-15263-7.  Google Scholar

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