  We investigate time-dependent mean-field games with superquadratic
Hamiltonians and a power dependence on the measure. Such problems pose
substantial mathematical challenges as the key techniques used in the
subquadratic case do not extend to the superquadratic setting. Because of the
superquadratic structure of the Hamiltonian, Lipschitz estimates for the
solutions of the Hamilton-Jacobi equation are obtained through a novel set of
techniques. These explore the parabolic nature of the problem through the
non-linear adjoint method. Well-posedness is proved by combining Lipschitz
regularity for the Hamilton-Jacobi equation with polynomial estimates for
solutions of the Fokker-Planck equation. Existence of classical solutions can
then be proved under conditions depending only on the growth of the Hamiltonian
and the dimension. Our results also add to the current understanding of
superquadratic Hamilton-Jacobi equations.
