  We study a family of reaction-diffusion equations that present a doubly
nonlinear character given by a combination of the $p$-Laplacian and the porous
medium operators. We consider the so-called slow diffusion regime,
corresponding to a degenerate behaviour at the level 0, \normalcolor in which
nonnegative solutions with compactly supported initial data have a compact
support for any later time. For some results we will also require $p\ge2$ to
avoid the possibility of a singular behaviour away from 0.
  Problems in this family have a unique (up to translations) travelling wave
with a finite front. When the initial datum is bounded, radially symmetric and
compactly supported, we will prove that solutions converging to 1 (which exist,
as we show, for all the reaction terms under consideration for wide classes of
initial data) do so by approaching a translation of this unique traveling wave
in the radial direction, but with a logarithmic correction in the position of
the front when the dimension is bigger than one. As a corollary we obtain the
asymptotic location of the free boundary and level sets in the non-radial case
up to an error term of size $O(1)$. In dimension one we extend our results to
cover the case of non-symmetric initial data, as well as the case of bounded
initial data with supporting sets unbounded in one direction of the real line.
A main technical tool of independent interest is an estimate for the flux.
  Most of our results are new even for the special cases of the porous medium
equation and the $p$-Laplacian evolution equation.
