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

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 $\underline{p}\le \bar{p}$ 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 $m\le p<\underline{p}$. 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. $q\ge 2$, is examined in [31].