Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Fujita-type blowup result and low energy scattering for a nonlinear
  Schr\"o\-din\-ger equation   In this paper we consider the nonlinear Schr\"o\-din\-ger equation $i u_t
+\Delta u +\kappa |u|^\alpha u=0$. We prove that if $\alpha <\frac {2} {N}$ and
$\Im \kappa <0$, then every nontrivial $H^1$-solution blows up in finite or
infinite time. In the case $\alpha >\frac {2} {N}$ and $\kappa \in {\mathbb
C}$, we improve the existing low energy scattering results in dimensions $N\ge
7$. More precisely, we prove that if $ \frac {8} {N + \sqrt{ N^2 +16N }} <
\alpha \le \frac {4} {N} $, then small data give rise to global, scattering
solutions in $H^1$.

16
The categories of this paper are: stat
18
The categories of this paper are: econ
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom