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

Abstract

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 $N\ge 2$, where ${\Delta }_{p}u=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$ 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.

27 pages