## Differential and Integral Equations

### The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^s$

#### Abstract

In this paper, we shall prove that the scattering operator for nonlinear Schr\"{o}dinger equations $iu_t+(-\Delta)^m u=\lambda_1|u|^{p_1}u+ \lambda_2|u|^{p_2}u$ carries a band $\dot B(p_1,p_2,\delta)$ in $H^s$ into $H^s$ for some $\delta>0$, where $\dot B(p_1,p_2,\delta)=\{\varphi\in H^s: \|\varphi\|_{\dot H^{s(p_1)}\cap\dot H^{s(p_2)}}\leq \delta\}$, $s(p_i)=n/2-2m/p_i$, $s(p_2)\leq s\le s(p_1)+1$, $4m/n\leq p_1\leq p_2\leq 4m/(n-2s)$.

#### Article information

Source
Differential Integral Equations, Volume 15, Number 9 (2002), 1073-1083.

Dates
First available in Project Euclid: 21 December 2012

Guo, Boling; Wang, Baoxiang. The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^s$. Differential Integral Equations 15 (2002), no. 9, 1073--1083. https://projecteuclid.org/euclid.die/1356060764