We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $\rho \left(x\right)$ and a power-like reaction term $u^p$ with $p>1$. The density decays fast at infinity, in the sense that $\rho \left(x\right)\sim {\left|x\right|}^{-q}$ as $\left|x\right|\to +\infty$ 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 slowly decaying density at infinity, i.e. $q\in [0,2)$, is examined in Meglioli and Punzo (2020).

我们关注密度可变的多孔介质方程的柯西问题的非负解 $\rho （X）$ 和类似幂的反应项 ${ü}^p$ 与 $p>\mathrm{1个}$。密度在无限远处快速衰减，$\rho （X）〜{\left|X\right|}^{-q}$ 如 $\left|X\right|\to +\infty$ 与 $q\ge 2。$ 在这种情况下 $q=2$如果 $p$ 大于 $米$，我们表明，对于足够大的初始数据，解决方案会在有限的时间内爆炸，而对于小的初始基准面，则存在全局解决方案。另一方面，当$q>2$，我们证明了全局及时解决方案的存在总是占主导地位。在无限远处缓慢衰减密度的情况，即$q\in [0，2）$，在Meglioli和Punzo（2020）中进行了研究。