Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Large time behavior for a porous medium equation in a nonhomogeneous
  medium with critical density   We study the large time behavior of solutions to the porous medium equation
in nonhomogeneous media with critical singular density $$
|x|^{-2}\partial_{t}u=\Delta u^m, \quad \hbox{in} \ \real^N\times(0,\infty), $$
where $m>1$ and $N\geq3$. The asymptotic behavior proves to have some
interesting and striking properties. We show that there are different
asymptotic profiles for the solutions, depending on whether the continuous
initial data $u_0$ vanishes at $x=0$ or not. Moreover, when $u_0(0)=0$, we show
the convergence towards a profile presenting a discontinuity in form of a
shockwave, coming from an unexpected asymptotic simplification to a
conservation law, while when $u_0(0)>0$, the limit profile remains continuous.
These phenomena illustrate the strong effect of the singularity at $x=0$. We
improve the time scale of the convergence in sets avoiding the singularity. On
the way, we also study the large-time behavior for a porous medium equation
with convection which is interesting for itself.

11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat