Time-dependent focusing Mean-Field Games: the sub-critical case

We consider time-dependent viscous mean-field games systems in the case of local, decreasing and unbounded couplings. These systems arise in mean-field game theory, and describe Nash equilibria of games with a large number of agents aiming at aggregation. We prove the existence of weak solutions that are minimizers of an associated non-convex functional, by rephrasing the problem in a convex framework. Under additional assumptions involving the growth at infinity of the coupling, the Hamiltonian, and the space dimension, we show that such minimizers are indeed classical solutions by a blow-up argument and additional Sobolev regularity for the Fokker–Planck equation. We exhibit an example of non-uniqueness of solutions. Finally, by means of a contraction principle, we observe that classical solutions exist just by local regularity of the coupling if the time horizon is short.