*Published Paper*

**Inserted:** 13 oct 2022

**Last Updated:** 13 oct 2022

**Year:** 2021

**Doi:** https://doi.org/10.1016/j.na.2020.112187

**Abstract:**

We are concerned with nonnegative solutions to the Cauchy problem for the
porous medium equation with a variable density $\rho(x)$ and a power-like
reaction term $u^p$ with $p>1$. The density decays {\it fast} at infinity, in
the sense that $\rho(x)\sim

x

^{-q}$ as $

x

\to +\infty$ with $q\ge 2.$ In the
case when $q=2$, if $p$ is bigger than $m$, we show that, for large enough
initial 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 show
that existence of global in time solutions always prevails. The case of {\it
slowly} decaying density at infinity, i.e. $q\in [0,2)$, is examined in 41.