  In this paper we investigate maximal $L^q$-regularity for time-dependent
viscous Hamilton-Jacobi equations with unbounded right-hand side and
superlinear growth in the gradient. Our approach is based on the interplay
between new integral and H\"older estimates, interpolation inequalities, and
parabolic regularity for linear equations. These estimates are obtained via a
duality method \`a la Evans. This sheds new light on a parabolic counterpart of
a conjecture by P.-L. Lions on maximal regularity for Hamilton-Jacobi
equations, recently addressed in the stationary framework by the authors.
Finally, applications to the existence problem of classical solutions to Mean
Field Games systems with unbounded local couplings are provided.
