Differential and Integral Equations

Time decay of small solutions to quadratic nonlinear Schrödinger equations in 3D

Nakao Hayashi, Tetsu Mizumachi, and Pavel I. Naumkin

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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 $\langle x\rangle =\sqrt{1+x^{2}}$.

Article information

Differential Integral Equations, Volume 16, Number 2 (2003), 159-179.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35B40: Asymptotic behavior of solutions 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein- Gordon and other equations of quantum mechanics


Hayashi, Nakao; Mizumachi, Tetsu; Naumkin, Pavel I. Time decay of small solutions to quadratic nonlinear Schrödinger equations in 3D. Differential Integral Equations 16 (2003), no. 2, 159--179. https://projecteuclid.org/euclid.die/1356060682

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