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

TL;DR

This paper studies the existence, uniqueness, and asymptotic behavior of positive solutions to a $p$-Laplacian parabolic equation with blow-up boundary conditions, considering a general absorption term with regular variation at infinity.

Contribution

It provides new results on the existence, uniqueness, and asymptotic analysis of solutions for a class of $p$-Laplacian parabolic equations with blow-up boundary conditions and general absorption functions.

Findings

Established conditions for existence and uniqueness of solutions.

Analyzed asymptotic behavior near the boundary.

Extended understanding of blow-up phenomena for $p$-Laplacian equations.

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 $N\ge2$, 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.