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

TL;DR

This paper investigates the conditions under which solutions to the porous medium equation with a reaction term and rapidly decaying density either blow up or exist globally, depending on the decay rate and initial data size.

Contribution

It provides new criteria for blow-up and global existence of solutions considering fast decaying densities and reaction effects, extending previous results.

Findings

Solutions blow up for large initial data when q=2 and p>m.

Solutions exist globally for all initial data when q>2.

The decay rate of density influences the long-term behavior of solutions.

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].