Existence of weak solutions to time-dependent mean-field games

Here, we establish the existence of weak solutions to a wide class time-dependent monotone mean-field games (MFGs). These MFGs are given as system degenerate parabolic equations with initial and terminal conditions. To construct these solutions, consider high-order elliptic regularization in space–time. Then, applying Schaefer’s fixed-point theorem, obtain uniqueness for this regularized problem. Using Minty’s method, prove solution original MFG. Finally, paper ends discussion on congestion problems density constrained MFGs.