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,\lambda u^\lambda,Du^\lambda)=F(x,m^\lambda(t))+c(m^\lambda(t)),\quad &&x\in M,\ \forall t\in[0,T],\ &\partial_t m^\lambda-\text{div}\left(m^\lambda\dfrac{\partial H}{\partial p}(x,\lambda u^\lambda,Du^\lambda)\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 Ma~n\'e 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.