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

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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 and 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.
[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.
[3]	R. Coifman and Y. Meyer, Ondelettes et Opérateurs. Ⅲ. Opérateurs Multilinéaires, Actualités Mathématiques. Hermann, Paris, 1991.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	N. Hayashi and P. Naumkin, Nongauge invariant cubic nonlinear Schrödinger equations, Pac. J. Appl. Math., 1 (2008), 1-16.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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
[20]	P. Naumkin, Cubic derivative nonlinear Schrödinger equations, SUT J. Math., 36 (2000), 9-42.
[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.
[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.
[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.
[24]	H. Sunagawa, Lower bounds of the lifespan of small data solutions to the nonlinear Schrödinger equations, Osaka J. Math., 43 (2006), 771-789.
[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.

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.
[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.
[3]	R. Coifman and Y. Meyer, Ondelettes et Opérateurs. Ⅲ. Opérateurs Multilinéaires, Actualités Mathématiques. Hermann, Paris, 1991.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	N. Hayashi and P. Naumkin, Nongauge invariant cubic nonlinear Schrödinger equations, Pac. J. Appl. Math., 1 (2008), 1-16.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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
[20]	P. Naumkin, Cubic derivative nonlinear Schrödinger equations, SUT J. Math., 36 (2000), 9-42.
[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.
[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.
[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.
[24]	H. Sunagawa, Lower bounds of the lifespan of small data solutions to the nonlinear Schrödinger equations, Osaka J. Math., 43 (2006), 771-789.
[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.

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