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On maximum and comparison principles for parabolic problems with the \(p\)-Laplacian. (English) Zbl 1416.35056
Summary: 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.

MSC:
35B50 Maximum principles in context of PDEs
35B51 Comparison principles in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35K20 Initial-boundary value problems for second-order parabolic equations
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