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

We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density $\rho (x)$ and a power-like reaction term. We show that for small enough initial data, if $\rho (x)\sim \frac{1}{{(\log \mathrm{\mid }x\mathrm{\mid })}^{\alpha }\mathrm{\mid }x{\mathrm{\mid }}^2}$ as $|x|\to \infty$,thensolutionsgloballyexistforany$p>1$\mathrm{.}Ontheotherhand,when$\rho(x)\sim\frac{\left(\log|x|\right)^{\alpha}}{|x|^{2}}$as$|x|\to \infty$, 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 $\rho (x)\sim \mathrm{\mid }x{\mathrm{\mid }}^{-q}$ for $q>0$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$.