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

## TL;DR

This paper studies the long-time behavior of solutions to a nonlinear p-Laplacian diffusion equation in tubular domains, revealing traveling wave phenomena and convergence to stationary profiles, with unique features due to nonlinear diffusion.

## Contribution

It demonstrates the existence of traveling wave solutions in logarithmic time for nonlinear diffusion in tubular domains and analyzes their role in the asymptotic behavior of solutions.

## Key findings

- Existence of traveling wave solutions connecting zero and steady states.
- Solutions converge to a universal stationary profile in the tube middle.
- Wave fronts exhibit Fisher-KPP type long-time behavior.

## 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.