Existence of Solutions and Selection Problem for Quasi-stationary Contact Mean Field Games

Abstract

First, we study the existence of solutions for a class of first order mean field games systems \begin{equation*} \left\{\begin{aligned} &H(x,u,Du)=F(x,m(t)),\quad &&x\in M,\ \forall\ t\in[0,T],\\ &\partial_t m-\text{div}\left(m\dfrac{\partial H}{\partial p}(x,u,Du)\right)=0,\quad &&(x,t)\in M\times(0,T],\\ &m(0)=m_0, \end{aligned}\right. \end{equation*} where the system comprises a stationary Hamilton-Jacobi equation in the contact case and an evolutionary continuity equation. Then, for any fixed $\lambda >0$, let $(u^{\lambda },m^{\lambda })$ be a solution of the system \begin{equation*} \left\{ \begin{aligned} &H(x,λu^λ,Du^λ)=F(x,m^λ(t))+c(m^λ(t)),\quad &&x\in M,\ \forall t\in[0,T],\\ &\partial_t m^λ-\text{div}\left(m^λ\dfrac{\partial H}{\partial p}(x,λu^λ,Du^λ)\right)=0,\quad &&(x,t)\in M\times(0,T],\\ &m(0)=m_0, \end{aligned}\right. \end{equation*} where $c(m^{\lambda }(t))$ is the Mañé critical value of the Hamiltonian $H(x,0,p)-F(x,m^{\lambda }(t))$. We investigate the selection problem for the limit of $(u^{\lambda },m^{\lambda })$ as $\lambda$ tends to 0.