Calculus of Variations and Geometric Measure Theory

G. Meglioli - F. Punzo

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

created by meglioli on 13 Oct 2022


Published Paper

Inserted: 13 oct 2022
Last Updated: 13 oct 2022

Year: 2021


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 $\rho(x)\sim
^{-q}$ as $
\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.