Short-Time Existence for a General Backward–Forward Parabolic System Arising from Mean-Field Games

  • Marco CirantEmail author
  • Roberto Gianni
  • Paola Mannucci


We study the local in time existence of a regular solution of a nonlinear parabolic backward–forward system arising from the theory of mean-field games (briefly MFG). The proof is based on a contraction argument in a suitable space that takes account of the peculiar structure of the system, which involves also a coupling at the final horizon. We apply the result to obtain existence to very general MFG models, including also congestion problems.


Parabolic equations Backward–forward system Mean-field games Hamilton–Jacobi Fokker–Planck Congestion problems 

Mathematics Subject Classification

35K40 35K61 49N90 



Marco Cirant and Paola Mannucci are members of GNAMPA-INdAM, and were partially supported by the research project of the University of Padova “Mean-Field Games and Nonlinear PDEs” and by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games”.


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Authors and Affiliations

  1. 1.Dipartimento di Matematica “Tullio Levi-Civita”Università di PadovaPaduaItaly
  2. 2.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFlorenceItaly

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