Title: Time-dependent focusing Mean-Field Games: the sub-critical case,
Abstract: We consider time-dependent viscous Mean-Field Games systems in the case of
local, decreasing and unbounded coupling. 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
minimisers 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 minimisers 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.