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$.