On maximum and comparison principles for parabolic problems with the $p$-Laplacian

TL;DR

This paper studies maximum and comparison principles for a class of quasilinear parabolic equations involving the p-Laplacian, providing conditions, counterexamples, and insights into their validity.

Contribution

It introduces new results and counterexamples for maximum and comparison principles in p-Laplacian parabolic problems, extending existing theory.

Findings

Established conditions for maximum principles

Presented counterexamples to certain assumptions

Analyzed the influence of parameters on principle validity

Abstract

We investigate strong and weak versions of maximum and comparison principles for a class of quasilinear parabolic equations with the $p$-Laplacian $$ \partial_t u - \Delta_p u = \lambda |u|^{p-2} u + f(x,t) $$ under zero boundary and nonnegative initial conditions on a bounded cylindrical domain $\Omega \times (0, T)$, $\lambda \in \mathbb{R}$, and $f \in L^\infty(\Omega \times (0, T))$. Several related counterexamples are given.