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2005 Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in 3D
Yuichiro Kawahara
Differential Integral Equations 18(2): 169-194 (2005).

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 $u_{0}.$}

Citation

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Yuichiro Kawahara. "Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in 3D." Differential Integral Equations 18 (2) 169 - 194, 2005.

Information

Published: 2005
First available in Project Euclid: 21 December 2012

zbMATH: 1212.35451
MathSciNet: MR2106101

Subjects:
Primary: 35Q55
Secondary: 35B40

Rights: Copyright © 2005 Khayyam Publishing, Inc.

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Vol.18 • No. 2 • 2005
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