  We study finite time blow-up and global existence of solutions to the Cauchy
problem for the porous medium equation with a variable density $\rho(x)$ and a
power-like reaction term. We show that for small enough initial data, if
$\rho(x)\sim \frac{1}{\left(\log|x|\right)^{\alpha}|x|^{2}}$ as $|x|\to
\infty$, then solutions globally exist for any $p>1$. On the other hand, when
$\rho(x)\sim\frac{\left(\log|x|\right)^{\alpha}}{|x|^{2}}$ as $|x|\to \infty$,
if the initial datum is small enough then one has global existence of the
solution for any $p>m$, while if the initial datum is large enough then the
blow-up of the solutions occurs for any $p>m$. Such results generalize those
established in [27] and [28], where it is supposed that $\rho(x)\sim |x|^{-q}$
for $q>0$ as $|x|\to \infty$.
