We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term up with p>1. The density decays fast at infinity, in the sense that ρ(x)∼|x|−q as |x|→+∞ with q≥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∈[0,2), is examined in Meglioli and Punzo (2020).
Blow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density
Giulia Meglioli;Fabio Punzo
2021-01-01
Abstract
We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term up with p>1. The density decays fast at infinity, in the sense that ρ(x)∼|x|−q as |x|→+∞ with q≥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∈[0,2), is examined in Meglioli and Punzo (2020).File | Dimensione | Formato | |
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