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 firstly consider the case that &rho;(x) decays at infinity like the critical case |x|<sup>-2</sup> 
divided by a positive power of the logarithm of |x| and we show that for small enough initial data, solutions globally exist for any p > 1. 
On the other hand, when &rho;(x) decays at infinity like the critical case |x|<sup>-2</sup> multiplied by a positive power of the logarithm of |x|, 
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) decays at infinity like a power of |x|, without logarithmic terms.