2003 Time decay of small solutions to quadratic nonlinear Schrödinger equations in 3D
Nakao Hayashi, Tetsu Mizumachi, Pavel I. Naumkin
Differential Integral Equations 16(2): 159-179 (2003). DOI: 10.57262/die/1356060682

Abstract

We study large time asymptotic behavior of small solutions to the quadratic nonlinear Schrödinger equation \begin{equation} \begin{cases} i\partial _{t}u+\frac{1}{2}\Delta u=\lambda u^{2}+\mu \overline{u}^{2},\text{ }(t,x)\in {\mathbf{R}}\times {\mathbf{R}}^{3}, \\ u(0,x)=u_{0}(x),\text{ }x\in {\mathbf{R}}^{3}, \end{cases} \tag*{(0.1)} \end{equation} where $\lambda$, $\mu\in\mathbf{C}$. We prove the $\mathbf{L}^{p}(2\leq p\leq \infty )$ - time decay estimate of solutions \begin{equation*} \big\| u ( t ) \big\| _{\mathbf{L}^{p}}\leq C\big( 1+ | t | \big) ^{-\frac{3}{2}\big( 1-\frac{2}{p}\big) }\big( \left\| u_{0}\right\| _{3,0}+\left\| u_{0}\right\| _{1,2}\big) \end{equation*} under the condition that $u_{0}\in \mathbf{H}^{3,0}\cap \mathbf{H}^{1,2}$ is sufficiently small, where \begin{equation*} \mathbf{H}^{m,k}=\left\{ \phi \in \mathbf{L}^{2}:\big\| \phi \big\| _{m,k}\equiv \big\| \langle x\rangle ^{k}\langle i\nabla \rangle ^{m}\phi \big\| _{\mathbf{L}^{2}} <\infty \right\} \end{equation*} is the weighted Sobolev space and $\langle x\rangle =\sqrt{1+x^{2}}$.

Citation

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Nakao Hayashi. Tetsu Mizumachi. Pavel I. Naumkin. "Time decay of small solutions to quadratic nonlinear Schrödinger equations in 3D." Differential Integral Equations 16 (2) 159 - 179, 2003. https://doi.org/10.57262/die/1356060682

Information

Published: 2003
First available in Project Euclid: 21 December 2012

zbMATH: 1031.35134
MathSciNet: MR1947090
Digital Object Identifier: 10.57262/die/1356060682

Subjects:
Primary: 35Q55
Secondary: 35B40 , 81Q05

Rights: Copyright © 2003 Khayyam Publishing, Inc.

Vol.16 • No. 2 • 2003
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