Properties of solutions to porous medium problems with different sources and boundary conditions

In this paper we study nonnegative and classical solutions $u=u(\text{nx},t)$ to porous medium problems of the type \begin{equation}\label{ProblemAbstract} \tag{$\mathrm{◊}$} \begin{cases} u_t=\Delta u^m + g(u,|\nabla u|) & {\bf x} \in \Omega, t\in I,\ %u_\nu+hu=0 & $\text{on}$; \partial \Omega, t>0,\ u({\bf x},0)=u_0({\bf x})&{\bf x} \in \Omega,\ \end{cases} \end{equation} where $\mathrm{\Omega }$ is a bounded and smooth domain of ${\mathbb{R}}^N$, with $N\ge 1$, $I=(0,t^{\ast })$ is the maximal interval of existence of $u$, $m>1$ and $u_0(\text{nx})$ is a nonngative and sufficiently regular function. The problem is equipped with different boundary conditions and depending on such boundary conditions as well as on the expression of the source $g$, global existence and blow-up criteria for solutions to \eqref{ProblemAbstract} are established. Additionally, in the three dimensional setting and when blow-up occurs, lower bounds for the blow-up time $t^{\ast }$ are also derived.