*Published Paper*

**Inserted:** 13 oct 2022

**Last Updated:** 13 oct 2022

**Year:** 2020

**Doi:** https://doi.org/10.1016/j.jde.2020.06.017

**Abstract:**

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 $q\in [0, 2).$ 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 $\underline p\leq \bar p$ depending on $\rho, m, N$, then the
solution blows-up in finite time for any nontrivial initial datum; we need the
extra hypotehsis that $q\in [0, \epsilon)$ for $\epsilon>0$ small enough, when
$m\leq p<\underline p$. 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. $q\geq 2$, is examined in 31.