  In this article, we investigate the initial and boundary blow-up problem for
the $p$-Laplacian parabolic equation $u_t-\Delta_p u=-b(x,t)f(u)$ over a smooth
bounded domain $\Omega$ of $\mathbb{R}^N$ with $N\ge2$, where $\Delta_pu={\rm
div}(|\nabla u|^{p-2}\nabla u)$ with $p>1$, and $f(u)$ is a function of regular
variation at infinity. We study the existence and uniqueness of positive
solutions, and their asymptotic behaviors near the parabolic boundary.
