Diffusion-aggregation processes with mono-stable reaction terms

This paper analyses front propagation of the aggregation-diffusion-reaction equation with a monostable reaction term and a diffusion coefficient which changes sign from positive to negative values. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density. The existence of infinitely many traveling wave solutions is proven. These fronts are parameterized by their wave speed and monotonically connect the stationary states 0 and 1. In the degenerate case sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both.