Summary: 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 [S. Kamin and P. Rosenau, J. Math. Phys. 23, 1385–1390 (1982; Zbl 0499.76111)]). 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 \(p > \overline{p}\), for a suitable \(\overline{p}\) depending on \(\rho, m, N\), then global solutions exist. In addition, if \(p < \underline p\), for a suitable \(\underline p \leq \overline{p}\) depending on \(\rho, m, N\), then the solution blows-up in finite time for any nontrivial initial datum; we need the extra hypothesis that \(q \in [0, \epsilon)\) for \(\epsilon > 0\) small enough, when \(m \leq p < \underline p\). Observe that \(\underline p = \overline{p}\), if \(\rho(x)\) is a multiple of \(| x |^{- q}\) for \(| x |\) large enough. Such results are in agreement with those established in [A. A. Samarskii et al., Blow-up in quasilinear parabolic equations. Transl. from the Russian by Michael Grinfeld. Berlin: Walter de Gruyter (1995; Zbl 1020.35001)], where \(\rho(x) \equiv 1\), and are related to some results in [A. V. Martynenko et al., “On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source”, Zh. Vychisl. Mat. Mat. Fiz. 48, No. 7, 1214–1229 (2008); Izv. Math. 76, No. 3, 563–580 (2012; Zbl 1255.35146); translation from Izv. Ross. Akad. Nauk, Ser. Mat 76, No. 3, 139–156 (2012)]. The case of fast decaying density at infinity, i.e. \(q \geq 2\), is examined in [the authors, J. Differ. Equations 269, No. 10, 8918–8958 (2020; Zbl 07216772)].