Blow-up and global existence for the inhomogeneous porous medium equation with reaction

2020/07/22 by Meglioli, Giulia, Punzo, Fabio
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.12005

Abstract

We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term. We show that for small enough initial data, if ρ(x)∼ \frac1(log|x|)^α|x|² as |x|→ ∞, then solutions globally exist for any p>1. On the other hand, when ρ(x)∼\frac(log|x|)^α|x|² as |x|→ ∞, if the initial datum is small enough then one has global existence of the solution for any p>m, while if the initial datum is large enough then the blow-up of the solutions occurs for any p>m. Such results generalize those established in [27] and [28], where it is supposed that ρ(x)∼ |x|^-q for q>0 as |x|→ ∞.

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