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>1p>1, on bounded cylindrical domains of RN+1RN+1,∂tu−Δpu−λ|u|p−2u≥0,∂tu−Δpu−λ|u|p−2u≥0, with nonnegative initial–boundary conditions and λ≤0λ≤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 12p>2. Also the Weak Maximum Principle for λ≤λ1λ≤λ1 is established.