Existence of solutions and selection problem for quasi-stationary contact mean field games

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 16 May 2025
Accepted: 5 February 2026

Abstract

First, we study the existence of solutions for a class of first order mean field games systems

H(x, u, D_u) = F (x, m(t)),      x ∈ M, ∀ t ∈ [0, T],

∂_tm - div (m ∂H/∂_p(x, u, Du) = 0,  (x, t) ∈ M × (0, T],

m(0) = m₀,

where the system comprises a stationary Hamilton-Jacobi equation in the contact case and an evolutionary continuity equation. Then, for any fixed λ > 0, let (u^λ,m^λ) be a solution of the system

H(x, λu^λ, Du^λ) = F (x, m^λ(t)) + c(m^λ(t)),  x ∈ M, ∀t ∈ [0, T],

∂_tm^λ - div (m^λ∂H/∂_p(x, λu^λ, Du^λ) = 0,   (x, t) ∈ M × (0, T],

m(0) = m₀,

where c(m^λ(t)) is the Mañé critical value of the Hamiltonian H(x, 0,p) — F(x,m^λ(t)). We investigate the selection problem for the limit of (u^λ, m^λ) as λ tends to 0.