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In this paper, we study the higher regularity theory of a mixed-typeparabolic problem. We extend the recent work of \cite{DMR} to constructsolutions that have an arbitrary number of derivatives in Sobolev spaces. Toachieve 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 tothe Prandtl system near the classical Falkner-Skan self-similar profiles inorder to supplement the stability analysis of \cite{IM22} with a rigorousconstruction argument.

Higher Regularity Theory for a Mixed-Type Parabolic Equation