• 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
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 & 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.  Google Scholar

[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.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Math., 10, Amer. Math. Soc., 2003. doi: 10.1090/cln/010.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

N. Hayashi, C. Li and P. I. Naumkin, Dissipative nonlinear Schrödinger equations with singular data, J. Appl. Computat. Math, 5 (2016), 1000304. Google Scholar

[8]

N. HayashiK. 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.  Google Scholar

[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.   Google Scholar

[10]

G. JinY. 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.  Google Scholar

[11]

S. KatayamaC. 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.   Google Scholar

[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.   Google Scholar

[13]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Phys. Théor., 46 (1987), 113-129.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

V. A. Liskevich and M. A. Perelmuter, Analyticity of submarkovian semigroups, Amer. Math. Soc., 123 (1995), 1097-1104.  doi: 10.2307/2160706.  Google Scholar

[21]

T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys., 139 (1991), 479-493.   Google Scholar

[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.  Google Scholar

[23]

Y. SagawaH. 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.   Google Scholar

[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.  Google Scholar

[25]

W. A. Strauss, Nonlinear scattering theory, Scattering Theory in Mathematical Physics, Reidel, Dordrecht, Holland, (1974), 53–78. Google Scholar

[26]

C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation: Self-focusing and wave collapse, Appl. Math. Sci., 139 Springer, 1999.  Google Scholar

[27]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial Ekvac., 30 (1987), 115-125.   Google Scholar

[28]

K. Yajima, Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys., 110 (1987), 415-426.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Math., 10, Amer. Math. Soc., 2003. doi: 10.1090/cln/010.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

N. Hayashi, C. Li and P. I. Naumkin, Dissipative nonlinear Schrödinger equations with singular data, J. Appl. Computat. Math, 5 (2016), 1000304. Google Scholar

[8]

N. HayashiK. 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.  Google Scholar

[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.   Google Scholar

[10]

G. JinY. 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.  Google Scholar

[11]

S. KatayamaC. 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.   Google Scholar

[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.   Google Scholar

[13]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Phys. Théor., 46 (1987), 113-129.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

V. A. Liskevich and M. A. Perelmuter, Analyticity of submarkovian semigroups, Amer. Math. Soc., 123 (1995), 1097-1104.  doi: 10.2307/2160706.  Google Scholar

[21]

T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys., 139 (1991), 479-493.   Google Scholar

[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.  Google Scholar

[23]

Y. SagawaH. 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.   Google Scholar

[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.  Google Scholar

[25]

W. A. Strauss, Nonlinear scattering theory, Scattering Theory in Mathematical Physics, Reidel, Dordrecht, Holland, (1974), 53–78. Google Scholar

[26]

C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation: Self-focusing and wave collapse, Appl. Math. Sci., 139 Springer, 1999.  Google Scholar

[27]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial Ekvac., 30 (1987), 115-125.   Google Scholar

[28]

K. Yajima, Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys., 110 (1987), 415-426.   Google Scholar

[1]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[2]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[3]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[4]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[5]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[6]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[7]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[8]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[9]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[10]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[11]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299

[12]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[13]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[14]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[15]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[16]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112

[17]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

[18]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[19]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[20]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (164)
  • HTML views (82)
  • Cited by (1)

Other articles
by authors

[Back to Top]