Long-Time Behaviour of Interaction Models on Riemannian Manifolds with Bounded Curvature

We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modelled by an interaction potential. We consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution’s support. The analytical results are supported by numerical simulations for the equation set up on the rotation group and the hyperbolic plane.