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

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.