Blow-up and global existence for the inhomogeneous porous medium equation with reaction

We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term. We firstly consider the case that ρ(x) decays at infinity like the critical case [x]-2divided 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 ρ(x) decays at infinity like the critical case [x]-2multiplied 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 ρ(x) decays at infinity like a power of [x], without logarithmic terms