| Contents |
| We discuss global existence results and the occurrence of blow up phenomena for the nonlinear parabolic problem involving the $p$--Laplace operator of the form
$$
\left\{\begin{array}{ll}\label{pbm1}
\partial_t u=\Delta_p u+F(t,x,u)&\mbox{in}\ \Omega\ \mbox{ for }\ t>0,\\
\sigma \partial_t u+|\nabla u|^{p-2}\partial_\nu u=0&\mbox{on}\
\partial\Omega\ \mbox{ for }\ t>0,\\
u(0,\cdot)=u_0 &\mbox{in}\ \overline{\Omega},\\
\end{array}\right.
$$
where $\Omega$ is a bounded domain of $\R^N$ with Lipschitz boundary, and where
$$\Delta_p u:=\div\, \Big(|\nabla u|^{p-2}\nabla u\Big)$$
is the $p$--\textit{Laplacian} defined for any $u\in W^{1,p}(\Omega)$ and $p>1$.
As for the \textit{dynamical} time lateral boundary condition $\sigma \partial_t u+|\nabla u|^{p-2}\partial_\nu u=0$
the coefficient $\sigma$ is assumed to be a nonnegative constant.
The parameter
dependent non linearity $F(\cdot,\cdot,u)=\lambda|u|^{p-2}u$ will be of particular interest, as well as the case $F(\cdot,\cdot,u)=\lambda|u|^{q-2}u$
with $q>p>2$.
The results presented here stem from a joint work with Joachim von Below and Ga\"elle Pincet Mailly. |
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