Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global existence for damped $\sigma$-evolution equations with nonlocal
  nonlinearity   In this research, we would like to study the global (in time) existence of
small data solutions to the following damped $\sigma$-evolution equations with
nonlocal (in space) nonlinearity: \begin{equation*}
\partial_{t}^{2}u+(-\Delta)^{\sigma}u+\partial_{t}u+(-\Delta)^{\sigma}\partial_{t}u=I_{\alpha}(|u|^{p}),
\ \ t>0, \ \ x\in \mathbb{R}^{n}, \end{equation*} where $\sigma\geq1$, $p>1$
and $I_{\alpha}$ is the Riesz potential of power nonlinearity $|u|^{p}$ for any
$\alpha\in (0,n)$. More precisely, by using the $(L^{m}\cap L^{2})-L^{2}$ and
$L^{2}-L^{2}$ linear estimates, where $m\in[1,2]$, we show the new influence of
the parameter $\alpha$ on the admissible ranges of the exponent $p$.

18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys