Local existence, global existence, and scattering for the nonlinear Schrödinger equation
Résumé
In this paper, we construct for every $\alpha >0$ and $\lambda \in {\mathbb C}$ a class of initial values for which there exists a local solution of
the nonlinear Schr\"o\-din\-ger equation
\begin{equation*}
\begin{cases}
iu_t + \Delta u + \lambda |u|^\alpha u= 0 \\ u(0,x) = u_0
\end{cases}
\end{equation*}
on ${\mathbb R}^N $. Moreover, we construct for every $\alpha >\frac {2} {N}$ a class of (arbitrarily large) initial values for which there exists a global solution that scatters as $t\to \infty $.