  We consider the parabolic, initial value problem $$ v_t =\Delta_p(v)+\lambda
g(x,v)\phi_p(v), \quad \text{in $\Omega \times (0,\infty),$} $$ \[ v =0,
\text{in $\partial\Omega \times (0,\infty),$}\tag{IVP} v =v_0\ge0, \text{in
$\Omega \times \{0\},$} \] where $\Omega$ is a bounded domain in ${\mathbb
R}^N$, for some integer $N\ge1$, with smooth boundary $\partial\Omega$,
$\phi_p(s):=|s|^{p-1} {\rm sgn}s$, $s\in{\mathbb R}$, $\Delta_p$ denotes the
$p$-Laplacian, with $p>\max\{2,N\}$, $v_0\in C^0(\overline{\Omega})$, and
$\lambda>0$. The function $g:\overline{\Omega } \times [0,\infty)\to(0,\infty)$
is $C^0$ and, for each $x\in\overline{\Omega }$, the function
$g(x,\cdot):[0,\infty)\to(0,\infty)$ is Lipschitz continuous and strictly
decreasing.
  Clearly, (IVP) has the trivial solution $v\equiv0$, for all $\lambda>0$. In
addition, there exists $0<\lambda_{\rm min}(g)<\lambda_{\rm max}(g)$
($\lambda_{\rm max}(g)$ may be $\infty$) such that:
  $(a)$ if $\lambda\not\in(\lambda_{\rm min}(g),\lambda_{\rm max}(g))$ then
(IVP) has no non-trivial, positive equilibrium;
  $(b)$ if $\lambda\in(\lambda_{\rm min}(g),\lambda_{\rm max}(g))$ then (IVP)
has a unique, non-trivial, positive equilibrium $e_\lambda\in
W_0^{1,p}(\Omega)$.
  We prove the following results on the positive solutions of (IVP):
  $(a)$ if $0<\lambda<\lambda_{\rm min}(g)$ then the trivial solution is
globally asymptotically stable;
  $(b)$ if $\lambda_{\rm min}(g)<\lambda<\lambda_{\rm max}(g)$ then $e_\lambda$
is globally asymptotically stable;
  $(c)$ if $\lambda_{\rm max}(g)<\lambda$ then any non-trivial solution blows
up in finite time.
