Mathematics – Analysis of PDEs
Scientific paper
2006-07-25
Mathematics
Analysis of PDEs
18 pages, no figures
Scientific paper
The initial value problem for the $L^{2}$ critical semilinear Schr\"odinger
equation in $\R^n, n \geq 3$ is considered. We show that the problem is
globally well posed in $H^{s}({\Bbb R^{n}})$ when $1>s>\frac{\sqrt{7}-1}{3}$
for $n=3$, and when $1>s> \frac{-(n-2)+\sqrt{(n-2)^2+8(n-2)}}{4}$ for $n \geq
4$. We use the ``$I$-method'' combined with a local in time Morawetz estimate.
Pavlović Nataša
Silva Daniela de
Staffilani Gigliola
Tzirakis Nikolaos
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