  Local well-posedness is established for a highly nonlocal nonlinear
diffusion-adhesion system for bounded initial values with small support.
Macroscopic systems of this kind were previously obtained by the authors
through upscaling in [32] and can account for the effect of microscopic
receptor binding dynamics in cell-cell adhesion. The system analysed here
couples an integro-PDE featuring degenerate diffusion of the porous media type
and nonlocal adhesion with a novel nonlinear integral equation. The approach is
based on decoupling the system and using Banach's fixed point theorem to solve
each of the two equations individually and subsequently the entire system. The
main challenge of the implementation lies in selecting a suitable framework.
One of the key results is the local well-posedness for the integral equation
with a Radon measure as a parameter. The analysis of this equation utilizes the
Kantorovich-Rubinstein norm, marking the first application of this norm in
handling a nonlinear integral equation.
