  For a fixed bounded domain $D \subset \mathbb{R}^N$ we investigate the
asymptotic behaviour for large times of solutions to the $p$-Laplacian
diffusion equation posed in a tubular domain \begin{equation*} \partial_t u =
\Delta_p u \quad \text{ in } D \times \mathbb{R}, \quad t > 0 \end{equation*}
with $p>2$, i.e., the slow diffusion case, and homogeneous Dirichlet boundary
conditions on the tube boundary. Passing to suitable re-scaled variables, we
show the existence of a travelling wave solution in logarithmic time that
connects the level $u = 0$ and the unique nonnegative steady state associated
to the re-scaled problem posed in a lower dimension, i.e. in $D\subset
\mathbb{R}^N$.
  We then employ this special wave to show that a wide class of solutions
converge to the universal stationary profile in the middle of the tube and at
the same time they spread in both axial tube directions, miming the behaviour
of the travelling wave (and its reflection) for large times.
  The first main feature of our analysis is that wave fronts are constructed
through a (nonstandard) combination of diffusion and absorbing boundary
conditions, which gives rise to a sort of Fisher-KPP long-time behaviour. The
second one is that the nonlinear diffusion term plays a crucial role in our
analysis. Actually, in the linear diffusion framework $p=2$ solutions behave
quite differently.
