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.