Global existence and blow-up of solutions to the porous medium equation with reaction and singular coefficients

Abstract

We study global in time existence versus blow-up in finite time of solutions to the Cauchy problem for the porous medium equation with a variable density $ \rho(x) $ and a power-like reaction term posed in the one dimensional interval $ (-R,R) $, $ R>0 $. Here the weight function is singular at the boundary of the domain $ (-R,R) $, indeed it is such that $ \rho(x)\sim (R-|x|)^{-q} $ as $ |x|\to R $, with $ q\ge0 $. We show a different behavior of solutions depending on the three cases when $ q>2 $, $ q = 2 $ and $ q<2 $.