Nonlocal bilinear equations. Equilibrium solutions and diffusive limit

This paper deals with the qualitative analysis of a class of bilinear systems of equations describing the dynamics of individuals undergoing kinetic (stochastic) interactions. A corresponding evolution problem is formulated in terms of integro-differential (nonlocal) system of equations. A general existence theory is provided. Under the assumption of periodic boundary conditions and the interaction rates expressed in terms of convolution operators two classes of equilibrium solutions are distinguished. The first class contains only constant functions and the second one contains some nonconstant functions. In the scalar case (one equation) under suitable scaling, related to the shrinking of interaction range of each individual, the limit to the corresponding "macroscopic" equation is studied. The limiting equation turns out to be the (nonlinear) porous medium equation.