Nonlocal Cross-Interaction Systems on Graphs: Nonquadratic Finslerian Structure and Nonlinear Mobilities

We study the evolution of a system of two species with nonlinear mobility and nonlocal interactions on a graph whose vertices are given by an arbitrary, positive measure. To this end, we extend a recently introduced 2-Wasserstein-type quasi-metric on generalized graphs, which is based on an upwind-interpolation, to the case of two-species systems, concave, nonlinear mobilities, and p ≠ 2. We provide a rigorous interpretation of the interaction system as a gradient flow in the Finslerian setting, arising from the new quasi-metric. 2010 Mathematics Subject Classification. 49J40 (Variational inequalities), 45G10 (Other nonlinear integral equations), 49J45 (Methods involving semicontinuity and convergence; relaxation), 28A33 (Spaces of measures, convergence of measures); 35B38 (Critical points of functionals in context of PDEs (e.g., energy functionals);