  Given a bounded domain $D \subset \mathbb{R}^N$ and $m > 1$, we study the
long-time behaviour of solutions to the Porous Medium equation (PME) posed in a
tube \[ \partial_tu = \Delta u^m \quad \text{ in } D \times \mathbb{R}, \quad t
> 0, \] with homogeneous Dirichlet boundary conditions on the boundary
$\partial D \times \mathbb{R}$ and suitable initial datum at $t=0$. In two
previous works, V\'azquez and Gilding & Goncerzewicz proved that a wide class
of solutions exhibit a traveling wave behaviour, when computed at a logarithmic
time-scale and suitably renormalized. In this paper, we show that, for large
times, solutions converge in relative error to the Friendly Giant, i.e., the
unique nonnegative solution to the PME posed in the section $D$ of the tube
(with homogeneous Dirichlet boundary conditions) having a special self-similar
form. In addition, sharp rates of convergence and uniform bounds for the
location of the free boundary of solutions are given.
