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arXiv · 1603.03204

Local existence, global existence, and scattering for the nonlinear Schr\"odinger equation

Thierry Cazenave·Ivan Naumkin
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Abstract

In this paper, we construct for every $\alpha >0$ and $\lambda \in {\mathbb C}$ a space of initial values for which there exists a local solution of the nonlinear Schr\"odinger 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 $.

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Thierry Cazenave, Ivan Naumkin. 2016-03-10. Local existence, global existence, and scattering for the nonlinear Schr\"odinger equation. https://doi.org/10.1142/s0219199716500383

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