Fully Coupled Nonlocal Quasilinear Forward-Backward Parabolic Equations Arising from Mean Field Games

In this paper, we study fully coupled nonlocal second order quasilinear forward-backward partial differential equations (FBPDEs), which arise from solution of the mean field game (MFG) suggested by Lasry and Lions [Japan. J. Math. 2 (2007), p. 237 (Remark iv)]. We show the existence of solutions $(u,m)\in C^{1+\frac{1}{4},2+\frac{1}{2}}([0,T]\times\mathbb{R}^n)\times C^{\frac{1}{2}}([0,T],\mathcal{P}_1(\mathbb{R}^n))$, and also the uniqueness under an additional monotonicity condition. Then, we improve the regularity of our weak solution $m$ to get a classical solution under appropriate regularity assumptions on coefficients. The FBPDEs can be used to investigate a system of mean field equations (MFEs), where the backward one is a Hamilton-Jacobi-Bellman equation and the forward one is a Fokker-Planck equation. Moreover, we prove a verification theorem and give an optimal strategy of the associated MFG via the solution of MFEs. Finally, we address the linear-quadratic problems.