Local existence, global existence, and scattering for the nonlinear Schrödinger equation

Abstract

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 $

On a class of solutions to the generalized KdV type equation

“…In the case considered here, the non-linearity is non-Lipschitz. Motivated by the results in [2] and using the smoothing effects of Kato type [4] we shall obtain the desired local well-posedness result for the IVP (GK) for a class of data satisfying (1.5).…”

Modified scattering for the critical nonlinear Schrödinger equation

“…a solution of (1.1) with a sufficiently small initial value (in some appropriate sense) is asymptotic as t → ∞ to a solution of the free Schrödinger equation. See [21,7,8,5,6,16,4]. On the other hand, if α ≤ 2 N , then low energy scattering cannot be expected, see [20,Theorem 3.2 and Example 3.3, p. 68] and [1].…”

Spatial behavior for NLS and applications to scattering

“…In this note, we review recent results [6,7] on the local Cauchy problem and the asymptotic behavior of solutions for the nonlinear Schrödinger equation…”

“…We now must estimate the last term on the right-hand side of (6). We cannot simply use Sobolev's embedding…”