  We study the intermediate asymptotic behavior of solutions to the first-order
mean field games system with a local coupling, when the initial density is a
compactly supported function on the real line, and the coupling is of power
type. Addressing a question that was left open in arXiv:2308.00314, we prove
that the solutions converge to the self-similar profile. We proceed by
analyzing a continuous rescaling of the solution, and identifying an
appropriate Lyapunov functional. We identify a critical value for the parameter
of the coupling, which determines the qualitative behavior of the functional,
and the well-posedness of the infinite horizon system. Accordingly, we also
establish, in the subcritical and critical cases, a second convergence result
which characterizes the behavior of the full solution as the time horizon
approaches infinity. We also prove the corresponding results for the mean field
planning problem. A large part of our analysis and methodology apply just as
well to arbitrary dimensions. As such, this work is a major step towards
settling these questions in the higher-dimensional setting.
