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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.

Si studia un'equazione ellittico-parabolica che descrive la distribuzione degli sforzi all'interno di un viscosimetro rotante riempito con un fluido di Bingham. Si prova l'esistenza e l'unicità delle soluzioni e alcuni risultati sulla continuità della frontiera solido-liquido. In this paper, we consider an elliptic-parabolic equation ruling the stress distribution inside a rotating viscometer filled with a Bingham fluid. Existence and uniqueness of the weak solution and some results concerning the continuity of the boundary of the solid region are proved.