Existence and non-existence for time-dependent mean field games with strong aggregation

We investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form -σ m^α -σmα , α ≥ 2/N α≥2/N , where m is the population density and N is the dimension of the state space. We prove the existence of solutions under the assumption that σ σ is small enough. For large σ σ , we show that existence may fail whenever the time horizon T is large.