International audienceIn 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 ofthe 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 $

Abstract. We consider the nonlinear Schrödinger equationwhere λ ∈ C and ℑλ ≤ 0. We construct a class of initial values for which the corresponding solution is global and decays as t → ∞, like t − N 2 if ℑλ = 0 and like (t log t) − N 2 if ℑλ < 0. Moreover, we give an asymptotic expansion of those solutions as t → ∞. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at u = 0. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.