Differential and Integral Equations

Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in 3D

Yuichiro Kawahara

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Abstract

We study the global existence and asymptotic behavior in time of small solutions to nonlinear Schrödinger equations with quadratic nonlinearities, \begin{eqnarray*} \left\{ \begin{array}{ll} i \partial_{t} u + \frac{1}{2}\Delta u = \mathcal{N}(u,\bar u), & (t, x) \in {\mbox{\bf{R}}} \times {\mbox{\bf{R}}}^3, \\ u(0, x) = u_{0}, & x \in {\mbox{\bf{R}}}^3, \end{array} \right. \label{0.1} \end{eqnarray*} where the initial data $u_{0}$ are sufficiently small in a suitable norm, $\bar{u}$ is the complex conjugate of $u$. The nonlinear term $\mathcal{N}$ is a smooth quadratic function in the neighborhood of the origin with respect to $u$ and $\bar u$ and does not contain the term $|u|^2$. Our purpose in this paper is to show there exists a unique final state $u_{+}$ such that $$ \big \|u(t)-e^{\frac{i t}{2} \Delta} u_{+} \big \|_{L^2} \le C t^{-\frac{5}{4}}, \ \ \text{ for small $u_{0}.$}

Article information

Source
Differential Integral Equations, Volume 18, Number 2 (2005), 169-194.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060228

Mathematical Reviews number (MathSciNet)
MR2106101

Zentralblatt MATH identifier
1212.35451

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35B40: Asymptotic behavior of solutions

Citation

Kawahara, Yuichiro. Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in 3D. Differential Integral Equations 18 (2005), no. 2, 169--194. https://projecteuclid.org/euclid.die/1356060228


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