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

Nakao Hayashi; Chunhua Li; Pavel I. Naumkin

doi:10.3934/cpaa.2017103

Article Contents

2017, Volume 16, Issue 6: 2089-2104. Doi: 10.3934/cpaa.2017103

This issue Previous Article Multiplicity results for fractional systems crossing high eigenvalues Next Article Multiple solutions for a fractional nonlinear Schrödinger equation with local potential

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
* Corresponding author

Received: November 30, 2016

Revised: April 30, 2017

Published: September 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B40.

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