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 $rho(x)$ and a power-like reaction term $rho(x) u^p$ with $p>1$; this is a mathematical model of a thermal evolution of a heated plasma (see [25]). The density decays slowly at infinity, in the sense that $rho(x)lesssim |x|^{-q}$ as $|x|to +infty$ with $qin [0, 2).$ 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 $p>bar p$, for a suitable $bar p$ depending on $rho, m, N$, then global solutions exist. In addition, if $p<underline p$, for a suitable $underline pleq bar p$ depending on $rho, m, N$, then the solution blows-up in finite time for any nontrivial initial datum; we need the extra hypotehsis that $qin [0, epsilon)$ for $epsilon>0$ small enough, when $mleq p<underline p$. Observe that $underline p=bar p$, if $rho(x)$ is a multiple of $|x|^{-q}$ for $|x|$ large enough. Such results are in agreement with those established in [41], where $rho(x)equiv 1$. The case of fast decaying density at infinity, i.e. $qgeq 2$, is examined in [31].
Download