  We consider reaction-diffusion equations of porous medium type, with
different kind of reaction terms, and nonnegative bounded initial data. For all
the reaction terms under consideration there are initial data for which the
solution converges to 1 uniformly in compact sets for large times. We will
characterize for which reaction terms this happens for all nontrivial
nonnegative initial data, and for which ones there are also solutions
converging uniformly to 0. Problems in this family have a unique (up to
translations) travelling wave with a finite front and we will see how its speed
gives the asymptotic velocity of all the solutions with compactly supported
initial data. We will also prove in the one-dimensional case that solutions
with bounded compactly supported initial data converging to 1 do so approaching
a translation of this unique traveling wave. We will prove a similar result for
non-compactly supported initial data in a certain class.
