# Time-dependent focusing Mean-Field Games: the sub-critical case

## TL;DR

This paper studies time-dependent Mean-Field Games with unbounded, decreasing coupling, proving existence of weak solutions, conditions for classical solutions, and demonstrating non-uniqueness, with implications for short time horizons.

## Contribution

It introduces a convex reformulation for existence proofs and provides regularity results under growth conditions, advancing understanding of solution behavior in complex MFG systems.

## Key findings

- Existence of weak solutions via convex reformulation.
- Conditions for classical solutions with regularity and growth assumptions.
- Non-uniqueness examples and short time horizon regularity results.

## Abstract

We consider time-dependent viscous Mean-Field Games systems in the case of local, decreasing and unbounded coupling. These systems arise in mean-field game theory, and describe Nash equilibria of games with a large number of agents aiming at aggregation. We prove the existence of weak solutions that are minimisers of an associated non-convex functional, by rephrasing the problem in a convex framework. Under additional assumptions involving the growth at infinity of the coupling, the Hamiltonian, and the space dimension, we show that such minimisers are indeed classical solutions by a blow-up argument and additional Sobolev regularity for the Fokker-Planck equation. We exhibit an example of non-uniqueness of solutions. Finally, by means of a contraction principle, we observe that classical solutions exist just by local regularity of the coupling if the time horizon is short.