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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
##### Keywords:
fast diffusion; slow diffusion; strong and weak versions
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##### References:
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