# Sharp profiles in models of collective movements

## TL;DR

This paper studies a non-monostable parabolic PDE model for crowd flows, focusing on semi-wavefront solutions, their properties, and convergence to wavefronts, revealing complex behaviors near the vanishing diffusivity and source points.

## Contribution

It introduces an analysis of semi-wavefront solutions in a non-monostable crowd flow model with vanishing diffusivity and source terms, exploring their existence, regularity, and convergence.

## Key findings

- Existence of semi-wavefront solutions
- Regularity and monotonicity properties established
- Convergence to wavefront solutions demonstrated

## Abstract

We consider a parabolic partial differential equation that can be understood as a simple model for crowds flows. Our main assumption is that the diffusivity and the source/sink term vanish at the same point; the nonhomogeneous term is different from zero at any other point and so the equation is not monostable. We investigate the existence, regularity and monotone properties of semi-wavefront solutions as well as their convergence to wavefront solutions.