Extinction in a finite time for solutions of a class of quasilinear parabolic equations

We study the property of extinction in a finite time for nonnegative solutions of [Formula: see text] for the Dirichlet Boundary Conditions when [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and Ω a bounded domain of [Formula: see text] ([Formula: see text]). We prove some necessary and sufficient conditions. The threshold is for power functions when [Formula: see text] while finite time extinction occurs for very flat potentials [Formula: see text] when [Formula: see text].