Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extinction of solutions to a class of fast diffusion systems with
  nonlinear sources   In this paper, the finite time extinction of solutions to the fast diffusion
system $u_t=\mathrm{div}(|\nabla u|^{p-2}\nabla u)+v^m$,
$v_t=\mathrm{div}(|\nabla v|^{q-2}\nabla v)+u^n$ is investigated, where
$1<p,q<2$, $m,n>0$ and $\Omega\subset \mathbb{R}^N\ (N\geq1)$ is a bounded
smooth domain. After establishing the local existence of weak solutions, the
authors show that if $mn>(p-1)(q-1)$, then any solution vanishes in finite time
provided that the initial data are ``comparable"; if $mn=(p-1)(q-1)$ and
$\Omega$ is suitably small, then the existence of extinction solutions for
small initial data is proved by using the De Giorgi iteration process and
comparison method. On the other hand, for $1<p=q<2$ and $mn<(p-1)^2$, the
existence of at least one non-extinction solution for any positive smooth
initial data is proved.

33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
16
The categories of this paper are: stat
18
The categories of this paper are: econ