  We deal with the large time behavior for a porous medium equation posed in
nonhomogeneous media with singular critical density $$
|x|^{-2}\partial_tu(x,t)=\Delta u^m(x,t), \quad (x,t)\in
\real^N\times(0,\infty), \ m\geq1, $$ posed in dimensions $N=1$ and $N=2$,
which are also interesting in applied models according to works by Kamin and
Rosenau. We deal with the Cauchy problem with bounded and continuous initial
data $u_0$. We show that in dimension $N=2$, the asymptotic profiles are
self-similar solutions that vary depending on whether $u_0(0)=0$ or
$u_0(0)=K\in(0,\infty)$. In dimension $N=1$, things are strikingly different,
and we find new asymptotic profiles of an unusual mixture between self-similar
and traveling wave forms. We thus complete the study performed in previous
recent works for the bigger dimensions $N\geq3$.
