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

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 نشر من قبل Giulia Meglioli
 تاريخ النشر 2019
  مجال البحث
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We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term $u^p$ with $p>1$. The density decays {it fast} at infinity, in the sense that $rho(x)sim |x|^{-q}$ as $|x|to +infty$ with $qge 2.$ 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 {it slowly} decaying density at infinity, i.e. $qin [0,2)$, is examined in [41].



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