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We consider a system of $n$ nonlocal interaction evolution equations on$\mathbb{R}^d$ with a differentiable matrix-valued interaction potential $W$.Under suitable conditions on convexity, symmetry and growth of $W$, we prove$\lambda$-geodesic convexity for some $\lambda\in\mathbb{R}$ of the associatedinteraction energy with respect to a weighted compound distance of Wassersteintype. In particular, this implies existence and uniqueness of solutions to theevolution system. In one spatial dimension, we further analyse the qualitativeproperties of this solution in the non-uniformly convex case. We obtain, if theinteraction potential is sufficiently convex far away from the origin, that thesupport of the solution is uniformly bounded. Under a suitable Lipschitzcondition for the potential, we can exclude finite-time blow-up and give apartial characterization of the long-time behaviour.

Geodesically convex energies and confinement of solutions for a multi-component system of nonlocal interaction equations