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 $-\sigma m^\alpha$, $\alpha \ge 2/N$, 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 $\sigma$ is small enough. For large $\sigma$, we show that existence may fail whenever the time horizon $T$ is large.