October  2016, 36(10): 5743-5761. doi: 10.3934/dcds.2016052

The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension

1. 

Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan, Japan

Received  October 2015 Revised  April 2016 Published  July 2016

Consider the initial value problem for cubic derivative nonlinear Schrödinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution, which can be computed explicitly from the initial data and the nonlinear term. This is an extension and a refinement of the previous work by one of the authors [H. Sunagawa: Osaka J. Math. 43 (2006), 771--789], in which the gauge-invariant nonlinearity was treated.
Citation: Yuji Sagawa, Hideaki Sunagawa. The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5743-5761. doi: 10.3934/dcds.2016052
References:
[1]

H. Chihara, Local existence for the semilinear Schrödinger equations in one space dimension, J. Math. Kyoto Univ., 34 (1994), 353-367.

[2]

H. Chihara, Gain of regularity for semilinear Schrödinger equations, Math. Ann., 315 (1999), 529-567. doi: 10.1007/s002080050328.

[3]

D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. doi: 10.1002/cpa.3160390205.

[4]

J.-M. Delort, Minoration du temps d'existence pour l'équation de Klein-Gordon non-linéaire en dimension 1 d'espace, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 563-591. doi: 10.1016/S0294-1449(99)80028-6.

[5]

J.-M. Delort, Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup.(4) 34 (2001), 1-61; Erratum: Ann. Sci. École Norm. Sup.(4) 39 (2006), 335-345. doi: 10.1016/S0012-9593(00)01059-4.

[6]

S. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions, J. Math. Kyoto Univ., 34 (1994), 319-328.

[7]

N. Hayashi and P. I. Naumkin, Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited, Discrete Contin. Dynam. Systems, 3 (1997), 383-400. doi: 10.3934/dcds.1997.3.383.

[8]

N. Hayashi and P. I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math., 120 (1998), 369-389. doi: 10.1353/ajm.1998.0011.

[9]

N. Hayashi and P. I. Naumkin, Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations without a self-conjugate property, Funkcial. Ekvac., 42 (1999), 311-324.

[10]

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

[11]

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

[12]

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

[13]

N. Hayashi and P. I. Naumkin, Asymptotics of odd solutions for cubic nonlinear Schrödinger equations, J. Differential Equations, 246 (2009), 1703-1722. doi: 10.1016/j.jde.2008.10.020.

[14]

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

[15]

N. Hayashi and P. I. Naumkin, Logarithmic time decay for the cubic nonlinear Schrödinger equations, Int. Math. Res. Not., IMRN 2015, 5604-5643. doi: 10.1093/imrn/rnu102.

[16]

N. Hayashi, P. I. Naumkin and P.-N. Pipolo, Smoothing effects for some derivative nonlinear Schrödinger equations, Discrete Contin. Dynam. Systems, 5 (1999), 685-695. doi: 10.3934/dcds.1999.5.685.

[17]

N. Hayashi, P. I. Naumkin and H. Sunagawa, On the Schrödinger equation with dissipative nonlinearities of derivative type, SIAM J. Math. Anal., 40 (2008), 278-291. doi: 10.1137/070689103.

[18]

N. Hayashi, P. I. Naumkin and H. Uchida, Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations, Publ. Res. Inst. Math. Sci., 35 (1999), 501-513. doi: 10.2977/prims/1195143611.

[19]

N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension, Differential Integral Equations, 7 (1994), 453-461.

[20]

L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Springer Lecture Notes in Math., 1256 (1987), 214-280. doi: 10.1007/BFb0077745.

[21]

L. Hörmander, Remarks on the Klein-Gordon equation, Journées équations aux derivées partielles (Saint Jean de Monts, 1-5 juin, 1987), Exp. No.I, 9pp., École Polytech., Palaiseau, 1987.

[22]

A. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014), 139-176. doi: 10.1016/j.jfa.2013.08.027.

[23]

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.

[24]

S. Katayama, A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ., 39 (1999), 203-213.

[25]

S. Katayama and Y. Tsutsumi, Global existence of solutions for nonlinear Schrödinger equations in one space dimension, Comm. Partial Differential Equations, 19 (1994), 1971-1997. doi: 10.1080/03605309408821079.

[26]

C. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255-288.

[27]

S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Applied Math., AMS, Providence, RI, 23 (1986), 293-326.

[28]

C. Li and H. Sunagawa, On Schrödinger systems with cubic dissipative nonlinearities of derivative type, Nonlinearity, 29 (2016), 1537-1563. doi: 10.1088/0951-7715/29/5/1537.

[29]

H. Lindblad and A. Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys., 73 (2005), 249-258. doi: 10.1007/s11005-005-0021-y.

[30]

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.

[31]

K. Moriyama, Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 10 (1997), 499-520.

[32]

K. Moriyama, S. Tonegawa and Y. Tsutsumi, Almost global existence of solutions for the quadratic semilinear Klein-Gordon equation in one space dimension, Funkcial. Ekvac., 40 (1997), 313-333.

[33]

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

[34]

P. I. Naumkin, The dissipative property of a cubic non-linear Schrödinger equation, Izv. Math., 79 (2015), 346-374. doi: 10.4213/im8179.

[35]

T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J., 45 (1996), 137-163. doi: 10.1512/iumj.1996.45.1962.

[36]

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

[37]

H. Sunagawa, Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan, 58 (2006), 379-400. doi: 10.2969/jmsj/1149166781.

[38]

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

[39]

S. Tonegawa, Global existence for a class of cubic nonlinear Schrödinger equations in one space dimension, Hokkaido Math. J., 30 (2001), 451-473. doi: 10.14492/hokmj/1350911962.

[40]

Y. Tsutsumi, The null gauge condition and the one-dimensional nonlinear Schrödinger equation with cubic nonlinearity, Indiana Univ. Math. J., 43 (1994), 241-254. doi: 10.1512/iumj.1994.43.43012.

[41]

S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135. doi: 10.1007/s00222-009-0176-8.

show all references

References:
[1]

H. Chihara, Local existence for the semilinear Schrödinger equations in one space dimension, J. Math. Kyoto Univ., 34 (1994), 353-367.

[2]

H. Chihara, Gain of regularity for semilinear Schrödinger equations, Math. Ann., 315 (1999), 529-567. doi: 10.1007/s002080050328.

[3]

D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. doi: 10.1002/cpa.3160390205.

[4]

J.-M. Delort, Minoration du temps d'existence pour l'équation de Klein-Gordon non-linéaire en dimension 1 d'espace, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 563-591. doi: 10.1016/S0294-1449(99)80028-6.

[5]

J.-M. Delort, Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup.(4) 34 (2001), 1-61; Erratum: Ann. Sci. École Norm. Sup.(4) 39 (2006), 335-345. doi: 10.1016/S0012-9593(00)01059-4.

[6]

S. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions, J. Math. Kyoto Univ., 34 (1994), 319-328.

[7]

N. Hayashi and P. I. Naumkin, Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited, Discrete Contin. Dynam. Systems, 3 (1997), 383-400. doi: 10.3934/dcds.1997.3.383.

[8]

N. Hayashi and P. I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math., 120 (1998), 369-389. doi: 10.1353/ajm.1998.0011.

[9]

N. Hayashi and P. I. Naumkin, Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations without a self-conjugate property, Funkcial. Ekvac., 42 (1999), 311-324.

[10]

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

[11]

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

[12]

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

[13]

N. Hayashi and P. I. Naumkin, Asymptotics of odd solutions for cubic nonlinear Schrödinger equations, J. Differential Equations, 246 (2009), 1703-1722. doi: 10.1016/j.jde.2008.10.020.

[14]

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

[15]

N. Hayashi and P. I. Naumkin, Logarithmic time decay for the cubic nonlinear Schrödinger equations, Int. Math. Res. Not., IMRN 2015, 5604-5643. doi: 10.1093/imrn/rnu102.

[16]

N. Hayashi, P. I. Naumkin and P.-N. Pipolo, Smoothing effects for some derivative nonlinear Schrödinger equations, Discrete Contin. Dynam. Systems, 5 (1999), 685-695. doi: 10.3934/dcds.1999.5.685.

[17]

N. Hayashi, P. I. Naumkin and H. Sunagawa, On the Schrödinger equation with dissipative nonlinearities of derivative type, SIAM J. Math. Anal., 40 (2008), 278-291. doi: 10.1137/070689103.

[18]

N. Hayashi, P. I. Naumkin and H. Uchida, Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations, Publ. Res. Inst. Math. Sci., 35 (1999), 501-513. doi: 10.2977/prims/1195143611.

[19]

N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension, Differential Integral Equations, 7 (1994), 453-461.

[20]

L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Springer Lecture Notes in Math., 1256 (1987), 214-280. doi: 10.1007/BFb0077745.

[21]

L. Hörmander, Remarks on the Klein-Gordon equation, Journées équations aux derivées partielles (Saint Jean de Monts, 1-5 juin, 1987), Exp. No.I, 9pp., École Polytech., Palaiseau, 1987.

[22]

A. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014), 139-176. doi: 10.1016/j.jfa.2013.08.027.

[23]

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.

[24]

S. Katayama, A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ., 39 (1999), 203-213.

[25]

S. Katayama and Y. Tsutsumi, Global existence of solutions for nonlinear Schrödinger equations in one space dimension, Comm. Partial Differential Equations, 19 (1994), 1971-1997. doi: 10.1080/03605309408821079.

[26]

C. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255-288.

[27]

S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Applied Math., AMS, Providence, RI, 23 (1986), 293-326.

[28]

C. Li and H. Sunagawa, On Schrödinger systems with cubic dissipative nonlinearities of derivative type, Nonlinearity, 29 (2016), 1537-1563. doi: 10.1088/0951-7715/29/5/1537.

[29]

H. Lindblad and A. Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys., 73 (2005), 249-258. doi: 10.1007/s11005-005-0021-y.

[30]

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.

[31]

K. Moriyama, Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 10 (1997), 499-520.

[32]

K. Moriyama, S. Tonegawa and Y. Tsutsumi, Almost global existence of solutions for the quadratic semilinear Klein-Gordon equation in one space dimension, Funkcial. Ekvac., 40 (1997), 313-333.

[33]

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

[34]

P. I. Naumkin, The dissipative property of a cubic non-linear Schrödinger equation, Izv. Math., 79 (2015), 346-374. doi: 10.4213/im8179.

[35]

T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J., 45 (1996), 137-163. doi: 10.1512/iumj.1996.45.1962.

[36]

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

[37]

H. Sunagawa, Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan, 58 (2006), 379-400. doi: 10.2969/jmsj/1149166781.

[38]

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

[39]

S. Tonegawa, Global existence for a class of cubic nonlinear Schrödinger equations in one space dimension, Hokkaido Math. J., 30 (2001), 451-473. doi: 10.14492/hokmj/1350911962.

[40]

Y. Tsutsumi, The null gauge condition and the one-dimensional nonlinear Schrödinger equation with cubic nonlinearity, Indiana Univ. Math. J., 43 (1994), 241-254. doi: 10.1512/iumj.1994.43.43012.

[41]

S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135. doi: 10.1007/s00222-009-0176-8.

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