American Institute of Mathematical Sciences

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 - A, 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. [2] H. Chihara, Gain of regularity for semilinear Schrödinger equations,, Math. Ann., 315 (1999), 529. doi: 10.1007/s002080050328. [3] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data,, Comm. Pure Appl. Math., 39 (1986), 267. 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. 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), 34 (2001), 1. 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. [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. 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. 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. [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. [11] N. Hayashi and P. I. Naumkin, Large time behavior for the cubic nonlinear Schrödinger equation,, Canad. J. Math., 54 (2002), 1065. 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. 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. 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. [15] N. Hayashi and P. I. Naumkin, Logarithmic time decay for the cubic nonlinear Schrödinger equations,, Int. Math. Res. Not., (2015), 5604. 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. 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. 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. 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. [20] L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations,, Springer Lecture Notes in Math., 1256 (1987), 214. doi: 10.1007/BFb0077745. [21] L. Hörmander, Remarks on the Klein-Gordon equation,, Journées équations aux derivées partielles (Saint Jean de Monts, (1987), 1. [22] A. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension,, J. Funct. Anal., 266 (2014), 139. 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. 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. [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. 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. [27] S. Klainerman, The null condition and global existence to nonlinear wave equations,, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, 23 (1986), 293. [28] C. Li and H. Sunagawa, On Schrödinger systems with cubic dissipative nonlinearities of derivative type,, Nonlinearity, 29 (2016), 1537. 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. 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. 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. [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. [33] P. I. Naumkin, Cubic derivative nonlinear Schrödinger equations,, SUT J. Math., 36 (2000), 9. [34] P. I. Naumkin, The dissipative property of a cubic non-linear Schrödinger equation,, Izv. Math., 79 (2015), 346. doi: 10.4213/im8179. [35] T. Ozawa, On the nonlinear Schrödinger equations of derivative type,, Indiana Univ. Math. J., 45 (1996), 137. 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. 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. 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. [39] S. Tonegawa, Global existence for a class of cubic nonlinear Schrödinger equations in one space dimension,, Hokkaido Math. J., 30 (2001), 451. 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. 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. doi: 10.1007/s00222-009-0176-8.

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References:
 [1] H. Chihara, Local existence for the semilinear Schrödinger equations in one space dimension,, J. Math. Kyoto Univ., 34 (1994), 353. [2] H. Chihara, Gain of regularity for semilinear Schrödinger equations,, Math. Ann., 315 (1999), 529. doi: 10.1007/s002080050328. [3] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data,, Comm. Pure Appl. Math., 39 (1986), 267. 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. 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), 34 (2001), 1. 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. [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. 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. 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. [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. [11] N. Hayashi and P. I. Naumkin, Large time behavior for the cubic nonlinear Schrödinger equation,, Canad. J. Math., 54 (2002), 1065. 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. 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. 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. [15] N. Hayashi and P. I. Naumkin, Logarithmic time decay for the cubic nonlinear Schrödinger equations,, Int. Math. Res. Not., (2015), 5604. 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. 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. 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. 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. [20] L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations,, Springer Lecture Notes in Math., 1256 (1987), 214. doi: 10.1007/BFb0077745. [21] L. Hörmander, Remarks on the Klein-Gordon equation,, Journées équations aux derivées partielles (Saint Jean de Monts, (1987), 1. [22] A. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension,, J. Funct. Anal., 266 (2014), 139. 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. 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. [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. 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. [27] S. Klainerman, The null condition and global existence to nonlinear wave equations,, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, 23 (1986), 293. [28] C. Li and H. Sunagawa, On Schrödinger systems with cubic dissipative nonlinearities of derivative type,, Nonlinearity, 29 (2016), 1537. 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. 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. 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. [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. [33] P. I. Naumkin, Cubic derivative nonlinear Schrödinger equations,, SUT J. Math., 36 (2000), 9. [34] P. I. Naumkin, The dissipative property of a cubic non-linear Schrödinger equation,, Izv. Math., 79 (2015), 346. doi: 10.4213/im8179. [35] T. Ozawa, On the nonlinear Schrödinger equations of derivative type,, Indiana Univ. Math. J., 45 (1996), 137. 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. 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. 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. [39] S. Tonegawa, Global existence for a class of cubic nonlinear Schrödinger equations in one space dimension,, Hokkaido Math. J., 30 (2001), 451. 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. 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. doi: 10.1007/s00222-009-0176-8.
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