Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density

We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term ρ(x)up with p>1; this is a mathematical model of a thermal evolution of a heated plasma (see [29]). The density decays slowly at infinity, in the sense that ρ(x)≲|x|−q as |x|→+∞ with q∈[0,2). We show that for large enough initial data, solutions blow-up in finite time for any p>1. On the other hand, if the initial datum is small enough and p>p¯, for a suitable p¯ depending on ρ,m,N, then global solutions exist. In addition, if p0 small enough, when m≤p