Models of collective movements with negative degenerate diffusivities

We consider an advection-diffusion equation whose diffusivity can be negative. This equation arises in the modeling of collective movements, where the negative diffusivity simulates an aggregation behavior. Under suitable conditions we prove the existence, uniqueness and qualitative properties of traveling-wave solutions connecting states where the diffusivity has opposite signs. These results are extended to end states where the diffusivity is positive but is negative in between. The vanishing-viscosity limit is also considered. Examples from real-world models are provided.