Porous medium flow in a tube. Travelling waves and KPP behaviour

We study the long-time behaviour of the solutions of the Porous Medium Equation ut = ∆u, m > 1, posed in a tube Ω = R × D, with D a bounded domain in R, and with homogeneous Dirichlet conditions on the lateral boundary. We show that the asymptotic behaviour of general nonnegative solutions follows the KPP pattern in suitable rescaled variables. We proceed as follows: we pass to the renormalized problem and show that this problem admits a wave solution that travels along the tube with constant speed with respect to the new time (which is logarithmic in the old time scale). This solution has a bounded free boundary as a forward front. We also show that the universal asymptotic pattern for solutions with compactly supported data is described in the renormalized form by two of these travelling waves going out in different directions, joined by a stationary profile in the middle region. We compare this situation with the heat equation case ut = ∆u that behaves quite differently.