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We are concerned with nonnegative solutions to the Cauchy problem for theporous medium equation with a variable density $\rho(x)$ and a power-likereaction term $u^p$ with $p>1$. The density decays {\it fast} at infinity, inthe sense that $\rho(x)\sim |x|^{-q}$ as $|x|\to +\infty$ with $q\ge 2.$ In thecase when $q=2$, if $p$ is bigger than $m$, we show that, for large enoughinitial data, solutions blow-up in finite time and for small initial datum,solutions globally exist. On the other hand, in the case when $q>2$, we showthat existence of global in time solutions always prevails. The case of {\itslowly} decaying density at infinity, i.e. $q\in [0,2)$, is examined in [41].

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