In this article, we investigate the parabolic logistic equation with blow-up initial and boundary values over a smooth bounded domain
$\Omega$:
\begin{align*}
   u_t-\Delta u=a(x,t)u-b(x,t)u^p\ \ \ \
 &{\rm in}\ \Omega\times(0,T),\\
 u=\infty\ \ \ \ &{\rm on}\ \partial\Omega\times(0,T)\cup\overline\Omega\times\{0\},
\end{align*}
where $T>0$ and $p>1$ are constants, $a$ and $b$ are continuous functions, with  $b>0$ in $\Omega\times [0,T)$, $b(x,T)\equiv 0$. We study the existence and uniqueness of positive solutions, and their asymptotic behavior near the parabolic boundary. Under the extra condition that  $b(x,t)\geq c(T-t)^\theta d(x,\partial\Omega)^\beta$ on $\Omega\times[0,T)$ for some constants $c>0$, $\theta>0$ and $\beta>-2$, we show that such a solution stays bounded in any compact subset of $\Omega$ as $t$ increases to $T$, and hence solves the equation up to $t=T$.