-
Previous Article
The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem
- CPAA Home
- This Issue
-
Next Article
Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term
Dissipative nonlinear schrödinger equations for large data in one space dimension
Tokyo Denki University, Division of Science, Hatoyama, Saitama, 350-0394, Japan |
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. $
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.
|
show all references
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.
|
[1] |
Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723 |
[2] |
Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2089-2104. doi: 10.3934/cpaa.2017103 |
[3] |
Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383 |
[4] |
Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024 |
[5] |
Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009 |
[6] |
Thierry Cazenave, Zheng Han. Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4801-4819. doi: 10.3934/dcds.2020202 |
[7] |
Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure and Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831 |
[8] |
Chunhua Li. Decay of solutions for a system of nonlinear Schrödinger equations in 2D. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4265-4285. doi: 10.3934/dcds.2012.32.4265 |
[9] |
Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93 |
[10] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 |
[11] |
Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073 |
[12] |
Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 |
[13] |
Mostafa Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont, Olivier Goubet. Discrete Schrödinger equations and dissipative dynamical systems. Communications on Pure and Applied Analysis, 2008, 7 (2) : 211-227. doi: 10.3934/cpaa.2008.7.211 |
[14] |
Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609 |
[15] |
Min Chen, Olivier Goubet. Long-time asymptotic behavior of dissipative Boussinesq systems. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 509-528. doi: 10.3934/dcds.2007.17.509 |
[16] |
Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475 |
[17] |
Thierry Colin, Pierre Fabrie. Semidiscretization in time for nonlinear Schrödinger-waves equations. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 671-690. doi: 10.3934/dcds.1998.4.671 |
[18] |
Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066 |
[19] |
Jaeyoung Byeon, Louis Jeanjean. Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 255-269. doi: 10.3934/dcds.2007.19.255 |
[20] |
Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2877-2891. doi: 10.3934/dcdss.2020456 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]