| Contents |
| We consider initial-boundary value problems for nonlinear parabolic equations with $p$-Laplacian
\begin{equation*}
\partial_t u - \Delta_p u - \lambda |u|^{p-2} u = f(t,x),
\end{equation*}
where $\lambda \in \mathbb{R}$ is a spectral parameter.
Problems of this type attract a lot of attention in the last decades.
This is due to the fact that solutions to such problems possesses many unusual qualitative effects, which are not observed in the linear case ($p = 2$), e.g., extinction in finite time, simultaneously backward, elliptic and forward Harnack's estimates, violation of the strong maximum principle and Hopf's lemma, etc.
In this talk we will speak about current state of affairs and recent progress in maximum and comparison principles for such kind of parabolic problems.
Some outstanding issues will be discussed. |
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