Abstract
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 RN+1,∂tu−Δpu−λ|u|p−2u≥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≤λ1 is established.
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