  In this paper we prove uniqueness in the inverse boundary value problem for
the three coefficient functions in the porous medium equation with an
absorption term $\epsilon\partial_t u-\nabla\cdot(\gamma\nabla u^m)+\lambda
u^q=0$, with $m>1$, $m^{-1}<q<\sqrt{m}$, with the space dimension 2 or higher.
This is a degenerate parabolic type quasilinear PDE which has been used as a
model for phenomena in fields such as gas flow (through a porous medium),
plasma physics, and population dynamics. In the case when $\gamma=1$ a priori,
we prove unique identifiability with data supported in an arbitrarily small
part of the boundary. Even for the global problem we improve on previous work
by obtaining uniqueness with a finite (rather than infinite) time of
observation and also by introducing the additional absorption term $\lambda
u^q$.
