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

TL;DR

This paper investigates the conditions under which solutions to a one-dimensional porous medium equation with a singular boundary weight either exist globally or blow up in finite time, depending on the singularity degree.

Contribution

It provides a detailed analysis of how the boundary singularity affects the global existence and blow-up behavior of solutions in a one-dimensional setting.

Findings

Solutions behave differently for q>2, q=2, and q<2.

The singularity degree q determines whether solutions exist globally or blow up.

The paper characterizes the solution behavior based on the boundary weight's singularity.

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 $q<2$.