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

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 [29]). The density decays slowly at infinity, in the sense that rho(x) less than or similar to vertical bar x vertical bar(-q) as vertical bar x vertical bar ->+ infinity with q is an element of [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) over bar for a suitable (p) over bar depending on rho, m, N, then global solutions exist. In addition if p < <(p)under bar> for a suitable (p) over bar <= (p) over bar depending on rho, m, N then the solution blows-up in finite time for any nontrivial initial datum; we need the extra hypothesis that q is an element of[0, epsilon) for epsilon > 0 small enough, when m <= p < <(p)under bar>. Observethat (p) under bar = (p) over bar, if rho(x) is a multiple of vertical bar x vertical bar(-q) for vertical bar x vertical bar large enough. Such results are in agreement with those established in [48], where rho(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]. (c) 2020 Elsevier Inc. All rights reserved.