# Semi-wavefront solutions in models of collective movements with density-dependent diffusivity

## TL;DR

This paper investigates semi-wavefront solutions in a scalar parabolic equation modeling collective movements with density-dependent diffusivity, establishing existence, properties, and constructing traveling wave solutions.

## Contribution

It introduces a novel approach to analyze semi-wavefront solutions in models with degenerate diffusion, extending the understanding of wave propagation in collective movement models.

## Key findings

- Existence of semi-wavefront solutions for all wave speeds
- Construction of traveling wave solutions from semi-wavefronts
- Properties of solutions analyzed using comparison techniques

## Abstract

This paper deals with a nonhomogeneous scalar parabolic equation with possibly degenerate diffusion term; the process has only one stationary state. The equation can be interpreted as modeling collective movements (crowd dynamics, for instance). We first prove the existence of semi-wavefront solutions for every wave speed; their properties are investigated. Then, a family of travelling wave solutions is constructed by a suitable combination of the previous semi-wavefront solutions. Proofs exploit comparison-type techniques and are carried out in the case of one spatial variable; the extension to the general case is straightforward.