*Published Paper*

**Inserted:** 13 oct 2022

**Last Updated:** 13 oct 2022

**Year:** 2021

**Abstract:**

We study finite time blow-up and global existence of solutions to the Cauchy
problem for the porous medium equation with a variable density $\rho(x)$ and a
power-like reaction term. We show that for small enough initial data, if
$\rho(x)\sim \frac{1}{\left(\log

x

\right)^{\alpha}

x

^{2}}$ as $

x

\to
\infty$, then solutions globally exist for any $p>1$. On the other hand, when
$\rho(x)\sim\frac{\left(\log

x

\right)^{\alpha}}{

x

^{2}}$ as $

x

\to \infty$,
if the initial datum is small enough then one has global existence of the
solution for any $p>m$, while if the initial datum is large enough then the
blow-up of the solutions occurs for any $p>m$. Such results generalize those
established in 27 and 28, where it is supposed that $\rho(x)\sim

x

^{-q}$
for $q>0$ as $

x

\to \infty$.