Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global existence and blowup for a class of the focusing nonlinear
  Schr\"odinger equation with inverse-square potential   We consider a class of the focusing nonlinear Schr\"odinger equation with
inverse-square potential \[ i\partial_t u + \Delta u -c|x|^{-2}u = - |u|^\alpha
u, \quad u(0)=u_0 \in H^1, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \]
where $d\geq 3$, $\frac{4}{d}\leq \alpha \leq \frac{4}{d-2}$ and $c\ne 0$
satisfies $c>-\lambda(d):=-\left(\frac{d-2}{2}\right)^2$. In the mass-critical
case $\alpha=\frac{4}{d}$, we prove the global existence and blowup below
ground states for the equation with $d\geq 3$ and $c>-\lambda(d)$. In the mass
and energy intercritical case $\frac{4}{d}<\alpha<\frac{4}{d-2}$, we prove the
global existence and blowup below the ground state threshold for the equation.
This extends similar results of \cite{KillipMurphyVisanZheng} and
\cite{LuMiaoMurphy} to any dimensions $d\geq 3$ and a full range
$c>-\lambda(d)$. We finally prove the blowup below ground states for the
equation in the energy-critical case $\alpha=\frac{4}{d-2}$ with $d\geq 3$ and
$c>-\frac{d^2+4d}{(d+2)^2} \lambda(d)$.

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The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th