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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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