| Contents |
| Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and $u$ be a
solution of
\begin{equation} \label{parabolicequation}
\left\{ \begin{array}{rcl} \frac{\partial}{\partial t}(u^q) - \nabla
(|\nabla u|^{p-2} \nabla u) + a(x) u^\lambda & = & 0\\ u(x,0) & = &
u_0(x) \end{array} \right.
\end{equation}
for the Dirichlet boundary condition where $0 < \lambda < q < p - 1$
and $a$ is nonnegative on $\Omega$. In what follows, the function
$a$ is degenerate, i.e., it vanishes on
sets of positive measure.\\
The point is : is there a finite time $T$ such that the solution $u$
vanishes in whole $\Omega$ ?\\
We will deal with some results comming from so-called semi-classical
methods. The first semi-classical method was initiated in 1997 by
V.A. Kondratiev and L. V\'eron, improved in 2001 by Y.B., B. Helffer
and L. V\'eron and optimized with A. Shishkov in 2007 for $q = 1$
and $p =
2$.\\
In 2001, some results have been established when $q = 1$ and $p > 2$
or when $q < 1$ and $p = 2$. We present some new results when $q \ne
1$ and $p \ne 2$. |
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