November  2017, 16(6): 2089-2104. doi: 10.3934/cpaa.2017103

Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations

1. 

Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043, Japan

2. 

Department of Mathematics, College of Science, Yanbian University, No. 977 Gongyuan Road, Yanji City, Jilin Province, 133002, China, and, Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico

* Corresponding author

Received  December 2016 Revised  May 2017 Published  July 2017

We study the upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations
$\begin{matrix} i{{\partial }_{t}}u+\frac{1}{2}\Delta u=\lambda {{\left| u \right|}^{p-1}}u,\left( t,x \right)\in {{\mathbb{R}}^{+}}\text{ }\!\!\times\!\!\text{ }{{\mathbb{R}}^{n}}, \\ u\left( 0,x \right)={{u}_{0}}\left( x \right),x\in {{\mathbb{R}}^{n}}, \\ \end{matrix}$
in space dimensions
$n=1,2$
or
$3$
, where
$\lambda =\lambda _{1}+i\lambda _{2},$
$\lambda _{j}∈ \mathbb{R},$
$j=1,2,$
$\lambda _{2}<0$
and the subcritical order of nonlinearity
$p=1+\frac{2}{n}-μ ,$
where
$μ >0$
is small enough.
Citation: Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2089-2104. doi: 10.3934/cpaa.2017103
References:
[1]

G. P. Agrawal, Nonlinear Fiber Optics, 2nd edition, Academic Press, Inc. , 1995. doi: 10.1007/3-540-46629-0_9. Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An introduction 2nd edition, Springer-Verlag, Berlin-N. Y. , 1976. doi: 10.1007/978-3-642-66451-9. Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. Google Scholar

[4]

J. GinibreT. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Annales de l'I.H.P. Physique théorique, 60 (1994), 211-239. Google Scholar

[5]

N. HayashiC. Li and P. I. Naumkin, Nonlinear Schrödinger systems in 2d with nondecaying final data, J. Differential Equations, 260 (2016), 1472-1495. doi: 10.1016/j.jde.2015.09.033. Google Scholar

[6]

N. Hayashi, C. Li and P. I. Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical fields Advances in Mathematical Physics 2016 (2016), 7 pages. doi: 10.1155/2016/3702738. Google Scholar

[7]

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

[8]

G. JinY. Jin and C. Li, The initial value problem for nonlinear Schödinger equations with a dissipative nonlinearity in one space dimension, J. Evolution Equations, 16 (2016), 983-995. doi: 10.1007/s00028-016-0327-5. Google Scholar

[9]

N. Kita and A. Shimomura, Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity, J. Differential Equations, 242 (2007), 192-210. doi: 10.1016/j.jde.2007.07.003. Google Scholar

[10]

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

[11]

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

[12]

J. -L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires Dunod-Gauthier-Villars, Paris, 1969. Google Scholar

show all references

References:
[1]

G. P. Agrawal, Nonlinear Fiber Optics, 2nd edition, Academic Press, Inc. , 1995. doi: 10.1007/3-540-46629-0_9. Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An introduction 2nd edition, Springer-Verlag, Berlin-N. Y. , 1976. doi: 10.1007/978-3-642-66451-9. Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. Google Scholar

[4]

J. GinibreT. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Annales de l'I.H.P. Physique théorique, 60 (1994), 211-239. Google Scholar

[5]

N. HayashiC. Li and P. I. Naumkin, Nonlinear Schrödinger systems in 2d with nondecaying final data, J. Differential Equations, 260 (2016), 1472-1495. doi: 10.1016/j.jde.2015.09.033. Google Scholar

[6]

N. Hayashi, C. Li and P. I. Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical fields Advances in Mathematical Physics 2016 (2016), 7 pages. doi: 10.1155/2016/3702738. Google Scholar

[7]

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

[8]

G. JinY. Jin and C. Li, The initial value problem for nonlinear Schödinger equations with a dissipative nonlinearity in one space dimension, J. Evolution Equations, 16 (2016), 983-995. doi: 10.1007/s00028-016-0327-5. Google Scholar

[9]

N. Kita and A. Shimomura, Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity, J. Differential Equations, 242 (2007), 192-210. doi: 10.1016/j.jde.2007.07.003. Google Scholar

[10]

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

[11]

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

[12]

J. -L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires Dunod-Gauthier-Villars, Paris, 1969. Google Scholar

[1]

Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701

[2]

Huijiang Zhao. Large time decay estimates of solutions of nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 69-114. doi: 10.3934/dcds.2002.8.69

[3]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15

[4]

Marcello D'Abbicco, Ruy Coimbra Charão, Cleverson Roberto da Luz. Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2419-2447. doi: 10.3934/dcds.2016.36.2419

[5]

Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic & Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605

[6]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135

[7]

Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic & Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251

[8]

Robert M. Strain, Keya Zhu. Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$. Kinetic & Related Models, 2012, 5 (2) : 383-415. doi: 10.3934/krm.2012.5.383

[9]

Youcef Mammeri. On the decay in time of solutions of some generalized regularized long waves equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 513-532. doi: 10.3934/cpaa.2008.7.513

[10]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481

[11]

Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013

[12]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-15. doi: 10.3934/dcds.2019229

[13]

Marco Cappiello, Fabio Nicola. Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1869-1880. doi: 10.3934/dcds.2016.36.1869

[14]

Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723

[15]

Linghai Zhang. Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1025-1042. doi: 10.3934/dcds.2002.8.1025

[16]

Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032

[17]

Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020009

[18]

Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164

[19]

Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407

[20]

Antonio Magaña, Alain Miranville, Ramón Quintanilla. On the time decay in phase–lag thermoelasticity with two temperatures. Electronic Research Archive, 2019, 27: 7-19. doi: 10.3934/era.2019007

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (15)
  • HTML views (8)
  • Cited by (0)

Other articles
by authors

[Back to Top]