# American Institute of Mathematical Sciences

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 and 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. [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. [3] T. Cazenave, Semilinear Schrödinger Equations Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. [4] J. Ginibre, T. 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. [5] N. Hayashi, C. 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. [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. [7] N. Hayashi, Li, Chunhua and P. I. Naumkin, Dissipative nonlinear Schrödinger equations with singular data, J. Appl. Computat. Math 5 (2016), 1000304. [8] G. Jin, Y. 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. [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. [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. [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. [12] J. -L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires Dunod-Gauthier-Villars, Paris, 1969.

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. [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. [3] T. Cazenave, Semilinear Schrödinger Equations Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. [4] J. Ginibre, T. 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. [5] N. Hayashi, C. 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. [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. [7] N. Hayashi, Li, Chunhua and P. I. Naumkin, Dissipative nonlinear Schrödinger equations with singular data, J. Appl. Computat. Math 5 (2016), 1000304. [8] G. Jin, Y. 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. [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. [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. [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. [12] J. -L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires Dunod-Gauthier-Villars, Paris, 1969.
 [1] Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon. Sharp large time behaviour in $N$-dimensional Fisher-KPP equations. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7265-7290. doi: 10.3934/dcds.2019303 [2] Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic and Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701 [3] Huijiang Zhao. Large time decay estimates of solutions of nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 69-114. doi: 10.3934/dcds.2002.8.69 [4] Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15 [5] 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 and Continuous Dynamical Systems, 2016, 36 (5) : 2419-2447. doi: 10.3934/dcds.2016.36.2419 [6] Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 [7] Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic and Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605 [8] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [9] David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135 [10] Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic and Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251 [11] Robert M. Strain, Keya Zhu. Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$. Kinetic and Related Models, 2012, 5 (2) : 383-415. doi: 10.3934/krm.2012.5.383 [12] Youcef Mammeri. On the decay in time of solutions of some generalized regularized long waves equations. Communications on Pure and Applied Analysis, 2008, 7 (3) : 513-532. doi: 10.3934/cpaa.2008.7.513 [13] Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic and Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481 [14] Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 [15] Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic and Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013 [16] Shihu Li, Wei Liu, Yingchao Xie. Small time asymptotics for SPDEs with locally monotone coefficients. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4801-4822. doi: 10.3934/dcdsb.2020127 [17] Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure and Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003 [18] Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3555-3577. doi: 10.3934/dcds.2021007 [19] Linghai Zhang. Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 1025-1042. doi: 10.3934/dcds.2002.8.1025 [20] Yuhui Chen, Ronghua Pan, Leilei Tong. The sharp time decay rate of the isentropic Navier-Stokes system in ${\mathop{\mathbb R\kern 0pt}\nolimits}^3$. Electronic Research Archive, 2021, 29 (2) : 1945-1967. doi: 10.3934/era.2020099

2021 Impact Factor: 1.273

## Metrics

• PDF downloads (178)
• HTML views (55)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]