# American Institute of Mathematical Sciences

October  2018, 11(5): 901-914. doi: 10.3934/dcdss.2018054

## One-dimensional, forward-forward mean-field games with congestion

 King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900. Saudi Arabia

* Corresponding author: Diogo Gomes

Received  March 2017 Revised  July 2017 Published  June 2018

Fund Project: D. Gomes was partially supported by KAUST baseline funds and KAUSTOSR-CRG2017-3452. M. Sedjro was supported by KAUST baseline and start-up funds

Here, we consider one-dimensional forward-forward mean-field games (MFGs) with congestion, which were introduced to approximate stationary MFGs. We use methods from the theory of conservation laws to examine the qualitative properties of these games. First, by computing Riemann invariants and corresponding invariant regions, we develop a method to prove lower bounds for the density. Next, by combining the lower bound with an entropy function, we prove the existence of global solutions for parabolic forward-forward MFGs. Finally, we construct traveling-wave solutions, which settles in a negative way the convergence problem for forward-forward MFGs. A similar technique gives the existence of time-periodic solutions for non-monotonic MFGs.

Citation: Diogo Gomes, Marc Sedjro. One-dimensional, forward-forward mean-field games with congestion. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 901-914. doi: 10.3934/dcdss.2018054
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##### References:
Domain $\mathcal{S}_1$
Level sets of the Riemann invariats for (37) with $\alpha = \frac 3 2$
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