We study the degenerate logistic model
described by the equation $ u_t - $Δ$ u=au-b(x)u^p$ with
standard boundary conditions, where $p>1$, $b$ vanishes on a
nontrivial subset $\Omega_0$ of the underlying bounded domain
$\Omega\subset R^N$ and $b$ is positive on
$\Omega_+=\Omega\setminus \overline{\Omega}_0$. We consider the
difficult case where $\partial\Omega_0\cap
\partial \Omega$≠$\emptyset$ and
$\partial\Omega_+\cap \partial \Omega$≠$\emptyset$, and
examine the asymptotic behaviour of the solutions. By a detailed
study of a singularly mixed boundary blow-up problem, we obtain
some basic results on the dynamics of the model.