We establish an intrinsic Harnack estimate for nonnegative weak solutions of the singular equation
(
1.1
)
(1.1)
below, for
m
m
in the optimal range
(
(
N
−
2
)
+
/
N
,
1
)
((N - 2)_+/N,1)
. Intrinsic means that, due to the singularity, the space-time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself. Consequences are, sharp supestimates on the solutions and decay rates as
t
t
approaches the extinction time. Analogous results are shown for
p
p
-laplacian type equations.