## Differential and Integral Equations

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

Yuichiro Kawahara

#### 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