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February  2020, 19(2): 967-981. doi: 10.3934/cpaa.2020044

## Dissipative nonlinear schrödinger equations for large data in one space dimension

 Tokyo Denki University, Division of Science, Hatoyama, Saitama, 350-0394, Japan

Received  February 2019 Revised  June 2019 Published  October 2019

In this study, we consider the global Cauchy problem for the nonlinear Schrödinger equations with a dissipative nonlinearity in one space dimension. In particular, we show the global existence, smoothing effect and asymptotic behavior for solutions to the nonlinear Schrödinger equations with data which belong to $\mathcal{F}H^\gamma,$ $1/2<\gamma\leq 1.$ In the proof of main theorem, we introduce a priori estimate for $H^\gamma$-type norm and the condition $\mathcal{F}H^1$ for data relaxed into $\mathcal{F}H^\gamma,$ $1/2<\gamma\leq1.$

Citation: Gaku Hoshino. Dissipative nonlinear schrödinger equations for large data in one space dimension. Communications on Pure and Applied Analysis, 2020, 19 (2) : 967-981. doi: 10.3934/cpaa.2020044
##### References:
 [1] H. Bahouri, J-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2011. doi: 10.1007/978-3-642-16830-7. [2] J. E. Barab, Nonexistence of asymptotically free solutions for nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.  doi: 10.1063/1.526074. [3] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Math., 10, Amer. Math. Soc., 2003. doi: 10.1090/cln/010. [4] J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimensions $n\geq2$, vCommun. Math. Phys., 151 (1993), 619-645. [5] J. Ginibre and G. Velo, On a class of nonlinear Schrodinger equations. Ⅰ: The Cauchy problem, J. Funct. Anal., 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4. [6] N. Hayashi, C. Li and P. I. Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical field, Adv. Math. Phys., (2016), 3702738. doi: 10.1155/2016/3702738. [7] N. Hayashi, C. Li and P. I. Naumkin, Dissipative nonlinear Schrödinger equations with singular data, J. Appl. Computat. Math, 5 (2016), 1000304. [8] N. Hayashi, K. Nakamitsu and M. Tsutsumi, On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension, Math. Z., 192 (1986), 637-650.  doi: 10.1007/BF01162710. [9] 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. [10] G. Jin, Y. Jin and C. Li, The initial value problem for nonlinear Schrödinger equations with a dissipative nonlinearity in one space dimension, J. Evol. Equ., 16 (2016), 983-995.  doi: 10.1007/s00028-016-0327-5. [11] S. Katayama, C. Li and H. Sunagawa, A remark on decay rates of solutions for a system of quadratic nonlinear Schrödinger equations in 2D, Differ. Int. Equ., 27 (2014), 310-312. [12] J. Kato and F. Pusateri, A new proof of long-range scattering for critical nonlinear Schrödinger equations, Differ. Int. Equ., 24 (2011), 923-940. [13] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Phys. Théor., 46 (1987), 113-129. [14] T. Kato, On nonlinear Schrödinger equations, Ⅱ. $H^s$-solutions and unconditional well-posedness, J. d'Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794. [15] N. Kita and A. Shimomura, Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity, J. Differ. Equ., 242 (2007), 192–210. doi: 10.1016/j.jde.2007.07.003. [16] N. Kita and A. Shimomura, Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data, J. Math. Soc. Japan, 61 (2009), 39-64. [17] C. Li and N. Hayashi, Critical nonlinear Schrödinger equations with data in homogeneous weighted $\mathbf{L}^2$ spaces, J. Math. Anal. Appl., 419 (2014), 1214-1234.  doi: 10.1016/j.jmaa.2014.05.053. [18] 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. [19] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edn., Springer, New York, 2015 doi: 10.1007/978-1-4939-2181-2. [20] V. A. Liskevich and M. A. Perelmuter, Analyticity of submarkovian semigroups, Amer. Math. Soc., 123 (1995), 1097-1104.  doi: 10.2307/2160706. [21] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys., 139 (1991), 479-493. [22] T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var., 25 (2006), 403-408.  doi: 10.1007/s00526-005-0349-2. [23] Y. Sagawa, H. Sunagawa and S. Yasuda, A sharp lower bound for the life span of small solutions to the Schödinger equation with a subcritical power nonlinearity, Differ. Int. Equ., 31 (2018), 685-700. [24] A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Commun. Partial Differ. Equ., 31 (2006), 1407-1423.  doi: 10.1080/03605300600910316. [25] W. A. Strauss, Nonlinear scattering theory, Scattering Theory in Mathematical Physics, Reidel, Dordrecht, Holland, (1974), 53–78. [26] C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation: Self-focusing and wave collapse, Appl. Math. Sci., 139 Springer, 1999. [27] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial Ekvac., 30 (1987), 115-125. [28] K. Yajima, Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys., 110 (1987), 415-426.

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##### References:
 [1] H. Bahouri, J-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2011. doi: 10.1007/978-3-642-16830-7. [2] J. E. Barab, Nonexistence of asymptotically free solutions for nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.  doi: 10.1063/1.526074. [3] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Math., 10, Amer. Math. Soc., 2003. doi: 10.1090/cln/010. [4] J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimensions $n\geq2$, vCommun. Math. Phys., 151 (1993), 619-645. [5] J. Ginibre and G. Velo, On a class of nonlinear Schrodinger equations. Ⅰ: The Cauchy problem, J. Funct. Anal., 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4. [6] N. Hayashi, C. Li and P. I. Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical field, Adv. Math. Phys., (2016), 3702738. doi: 10.1155/2016/3702738. [7] N. Hayashi, C. Li and P. I. Naumkin, Dissipative nonlinear Schrödinger equations with singular data, J. Appl. Computat. Math, 5 (2016), 1000304. [8] N. Hayashi, K. Nakamitsu and M. Tsutsumi, On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension, Math. Z., 192 (1986), 637-650.  doi: 10.1007/BF01162710. [9] 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. [10] G. Jin, Y. Jin and C. Li, The initial value problem for nonlinear Schrödinger equations with a dissipative nonlinearity in one space dimension, J. Evol. Equ., 16 (2016), 983-995.  doi: 10.1007/s00028-016-0327-5. [11] S. Katayama, C. Li and H. Sunagawa, A remark on decay rates of solutions for a system of quadratic nonlinear Schrödinger equations in 2D, Differ. Int. Equ., 27 (2014), 310-312. [12] J. Kato and F. Pusateri, A new proof of long-range scattering for critical nonlinear Schrödinger equations, Differ. Int. Equ., 24 (2011), 923-940. [13] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Phys. Théor., 46 (1987), 113-129. [14] T. Kato, On nonlinear Schrödinger equations, Ⅱ. $H^s$-solutions and unconditional well-posedness, J. d'Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794. [15] N. Kita and A. Shimomura, Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity, J. Differ. Equ., 242 (2007), 192–210. doi: 10.1016/j.jde.2007.07.003. [16] N. Kita and A. Shimomura, Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data, J. Math. Soc. Japan, 61 (2009), 39-64. [17] C. Li and N. Hayashi, Critical nonlinear Schrödinger equations with data in homogeneous weighted $\mathbf{L}^2$ spaces, J. Math. Anal. Appl., 419 (2014), 1214-1234.  doi: 10.1016/j.jmaa.2014.05.053. [18] 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. [19] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edn., Springer, New York, 2015 doi: 10.1007/978-1-4939-2181-2. [20] V. A. Liskevich and M. A. Perelmuter, Analyticity of submarkovian semigroups, Amer. Math. Soc., 123 (1995), 1097-1104.  doi: 10.2307/2160706. [21] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys., 139 (1991), 479-493. [22] T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var., 25 (2006), 403-408.  doi: 10.1007/s00526-005-0349-2. [23] Y. Sagawa, H. Sunagawa and S. Yasuda, A sharp lower bound for the life span of small solutions to the Schödinger equation with a subcritical power nonlinearity, Differ. Int. Equ., 31 (2018), 685-700. [24] A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Commun. Partial Differ. Equ., 31 (2006), 1407-1423.  doi: 10.1080/03605300600910316. [25] W. A. Strauss, Nonlinear scattering theory, Scattering Theory in Mathematical Physics, Reidel, Dordrecht, Holland, (1974), 53–78. [26] C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation: Self-focusing and wave collapse, Appl. Math. Sci., 139 Springer, 1999. [27] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial Ekvac., 30 (1987), 115-125. [28] K. Yajima, Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys., 110 (1987), 415-426.
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