Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Study of semi-linear $\sigma$-evolution equations with frictional and
  visco-elastic damping   In this article, we study semi-linear $\sigma$-evolution equations with
double damping including frictional and visco-elastic damping for any
$\sigma\ge 1$. We are interested in investigating not only higher order
asymptotic expansions of solutions but also diffusion phenomenon in the
$L^p-L^q$ framework, with $1\le p\le q\le \infty$, to the corresponding linear
equations. By assuming additional $L^{m}$ regularity on the initial data, with
$m\in [1,2)$, we prove the global (in time) existence of small data energy
solutions and indicate the large time behavior of the global obtained solutions
as well to semi-linear equations. Moreover, we also determine the so-called
critical exponent when $\sigma$ is integers.

29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
17
The categories of this paper are: q-fin
20
The categories of this paper are: acc-phys