Spatial behavior for NLS and applications to scattering
Séminaire Laurent Schwartz — EDP et applications (2017-2018), Talk no. 1, 11 p.

We review recent results on the nonlinear Schrödinger equation

iut+Δu+λ|u|αu=0

where λ and α>0. In any space dimension N1 and for any α>0, we construct a class of (arbitrarily large) initial values for which there exists a local solution. Moreover, if α>2/N, we construct a class of (arbitrarily large) initial values for which there exists a global solution that scatters as t. If α=2/N and λ0, we construct a class of (arbitrarily large) initial values for which there exists a global solution, of which we give a precise asymptotic expansion as t (of modified scattering type). These results rely on the construction of 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. This yields the desired asymptotic behavior if α>2/N. In the case α=2/N, a further step is required, and we estimate the solutions by allowing a certain growth of the Sobolev norms, which depends on the order of regularity through a cascade of exponents.

Published online:
DOI: 10.5802/slsedp.116
Cazenave, Thierry 1; Naumkin, Ivan 2

1 Sorbonne Université & CNRS, Laboratoire Jacques-Louis Lions B.C. 187 4 place Jussieu 75252 Paris Cedex 05 France
2 Laboratoire J.A. Dieudonné, UMR CNRS 7351, Université de Nice Sophia-Antipolis Parc Valrose 06108 Nice Cedex 02 France
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Cazenave, Thierry; Naumkin, Ivan. Spatial behavior for NLS and applications to scattering. Séminaire Laurent Schwartz — EDP et applications (2017-2018), Talk no. 1, 11 p. doi : 10.5802/slsedp.116. http://archive.numdam.org/articles/10.5802/slsedp.116/

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