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 ρ(x) and a power-like reaction term ρ(x)up with p>1; this is a mathematical model of a thermal evolution of a heated plasma (see [29]). The density decays slowly at infinity, in the sense that ρ(x)≲|x|−q as |x|→+∞ with q∈[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>p¯, for a suitable p¯ depending on ρ,m,N, then global solutions exist. In addition, if p<p_, for a suitable p_≤p¯ depending on ρ,m,N, then the solution blows-up in finite time for any nontrivial initial datum; we need the extra hypothesis that q∈[0,ϵ) for ϵ>0 small enough, when m≤p<p_. Observe that p_=p‾, if ρ(x) is a multiple of |x|−q for |x| large enough. Such results are in agreement with those established in [48], where ρ(x)≡1, and are related to some results in [32,33]. The case of fast decaying density at infinity, i.e. q≥2, is examined in [36].