A Strong Maximum Principle for parabolic equations with the p-Laplacian

We prove the Strong Maximum Principle (SMP) under suitable assumptions for a class of quasilinear parabolic problems with the p-Laplacian, p > 1 , on bounded cylindrical domains of R N + 1 , ∂ t u − Δ p u − λ | u | p − 2 u ≥ 0 , with nonnegative initial–boundary conditions and λ ≤ 0 , and we give some counterexamples to the SMP if some of our assumptions are violated. We show that the Hopf Maximum Principle holds for 1 < p < 2 , and give a counterexample to it for p > 2 . Also the Weak Maximum Principle for λ ≤ λ 1 is established.