  In this paper, we study the higher regularity theory of a mixed-type
parabolic problem. We extend the recent work of \cite{DMR} to construct
solutions that have an arbitrary number of derivatives in Sobolev spaces. To
achieve this, we introduce a counting argument based on a quantity called the
"degree". In the second part of this paper, we apply this existence theory to
the Prandtl system near the classical Falkner-Skan self-similar profiles in
order to supplement the stability analysis of \cite{IM22} with a rigorous
construction argument.
