Dissipative nonlinear schrödinger equations for large data in one space dimension

Gaku Hoshino

doi:10.3934/cpaa.2020044

Article Contents

2020, Volume 19, Issue 2: 967-981. Doi: 10.3934/cpaa.2020044

This issue Previous Article Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term Next Article The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem

Dissipative nonlinear schrödinger equations for large data in one space dimension

Gaku Hoshino

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

Received: January 31, 2019

Revised: May 31, 2019

Published: October 2019

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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. $

Keywords:

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q40.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.