  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.
