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

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.

16 pages