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

Abstract We investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings form $$-\sigma m^\alpha $$ -σmα , $$\alpha \ge 2/N$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">α≥2/N where m is population density N dimension state space. prove under assumption that $$\sigma xmlns:mml="http://www.w3.org/1998/Math/MathML">σ small enough. For large we show may fail whenever time horizon T large.