In this paper, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive p-Laplace equation

, 1 < p < 2, in a bounded domain

with

. More precisely, it is shown that if q > p-1, any solution vanishes in finite time when the initial datum or the coefficient a or the Lebesgue measure of the domain is small, and if 0 < q < p-1, there exists a solution which is positive in

for all t > 0. For the critical case q = p-1, whether the solutions vanish in finite time or not depends crucially on the value of

, where

and

is the unique positive solution of the elliptic problem -div(

) = 1,

;

=0,

. This is a main difference between equations with local and nonlocal sources.