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Multiple solutions for a fractional nonlinear Schrödinger equation with local potential
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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 |
| $\begin{matrix} i{{\partial }_{t}}u+\frac{1}{2}\Delta u=\lambda {{\left| u \right|}^{p-1}}u,\left( t,x \right)\in {{\mathbb{R}}^{+}}\text{ }\!\!\times\!\!\text{ }{{\mathbb{R}}^{n}}, \\ u\left( 0,x \right)={{u}_{0}}\left( x \right),x\in {{\mathbb{R}}^{n}}, \\ \end{matrix}$ |
| $n=1,2$ |
| $3$ |
| $\lambda =\lambda _{1}+i\lambda _{2},$ |
| $\lambda _{j}∈ \mathbb{R},$ |
| $j=1,2,$ |
| $\lambda _{2}<0$ |
| $p=1+\frac{2}{n}-μ ,$ |
| $μ >0$ |
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. |
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