  In this paper we analyze the porous medium equation
\begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} %\begin{cases}
u_t=\Delta u^m + a\io u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad
\textrm{in}\quad \Omega \times I,%\\ %u_\nu-g(u)=0 & \textrm{on}\; \partial
\Omega, t>0,\\ %u({\bf x},0)=u_0({\bf x})&{\bf x} \in \Omega,\\ %\end{cases}
\end{equation} where $\Omega$ is a bounded and smooth domain of $\R^N$, with
$N\geq 1$, and $I= [0,t^*)$ is the maximal interval of existence for $u$. The
constants $a,b,c$ are positive, $m,p,q$ proper real numbers larger than 1 and
the equation is complemented with nonlinear boundary conditions involving the
outward normal derivative of $u$. Under some hypothesis on the data, including
intrinsic relations between $m,p$ and $q$, and assuming that for some positive
and sufficiently regular function $u_0(\nx)$ the Initial Boundary Value Problem
(IBVP) associated to \eqref{ProblemAbstract} possesses a positive classical
solution $u=u(\nx,t)$ on $\Omega \times I$: \begin{itemize} \item
[$\triangleright$] when $p>q$ and in 2- and 3-dimensional domains, we determine
a \textit{lower bound of} $t^*$ for those $u$ becoming unbounded in
$L^{m(p-1)}(\Omega)$ at such $t^*$; \item [$\triangleright$] when $p<q$ and in
$N$-dimensional settings, we establish a \textit{global existence criterion}
for $u$. \end{itemize}
