Initial and boundary blow-up problem for $p$-Laplacian parabolic equation with general absorption

In this article, we investigate the initial and boundary blow-up problem for the $p$-Laplacian parabolic equation $u_t-{\mathrm{\Delta }}_{p}u=-b(x,t)f(u)$ over a smooth bounded domain $\mathrm{\Omega }$ of ${\mathbb{R}}^N$ with $N\ge 2$, 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.