Travelling wave behaviour arising in nonlinear diffusion problems posed in tubular domains

Abstract

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.