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

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

Nakao Hayashi 1,, , Chunhua Li 2, and  Pavel I. Naumkin 2,

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.
Keywords: Dissipative NLS equations, sharp time decay, large time asymptotics.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B40.
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. 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.   Google Scholar

[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.  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. 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.  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. 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.   Google Scholar

[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.  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. 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.  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

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