Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: System of Porous Medium Equations   We investigate the evolution of population density vector,
$\bold{u}=\left(u^1,\cdots,u^k\right)$, of $k$-species whose diffusion is
controlled by its absolute value $\left|\bold{u}\right|$. More precisely we
study the properties and asymptotic large time behaviour of solution
$\bold{u}=\left(u^1,\cdots,u^k\right)$ of degenerate parabolic system
\begin{equation*}
\left(u^i\right)_t=\nabla\cdot\left(\left|\bold{u}\right|^{m-1}\nabla
u^i\right) \qquad \mbox{for $m>1$ and $i=1,\cdots,k$}. \end{equation*} Under
some regularity assumption, we prove that the function $u^i$ which describes
the population density of $i$-th species with population $M_i$ converges to
$\frac{M_i}{\left|\bold{M}\right|}\mathcal{B}_{\left|\bold{M}\right|}$ in space
with two different approaches where $\mathcal{B}_{\left|\bold{M}\right|}$ is
the Barenblatt solution of the porous medium equation with $L^1$-mass
$\left|\bold{M}\right|=\sqrt{M_1^2+\cdots+M_k^2}$. \indent As an application of
the asymptotic behaviour, we establish a suitable harnack type inequality which
makes the spatial average of $u^i$ under control by the value of $u^i$ at one
point. We also find an 1-directional travelling wave type solutions and the
properties of solutions which has travelling wave behaviour at infinity.

16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th