  Supersonic-sonic patches are ubiquitous in regions of transonic flows and
they boil down to a family of degenerate hyperbolic problems in regions
surrounded by a streamline, a characteristic curve and a possible sonic curve.
This paper establishes the global existence of solutions in a whole
supersonic-sonic patch characterized by the two-dimensional full system of
steady Euler equations and studies solution behaviors near sonic curves,
depending on the proper choice of boundary data extracted from the airfoil
problem and related contexts. New characteristic decompositions are developed
for the full system and a delicate local partial hodograph transformation is
introduced for the solution estimates. It is shown that the solution is
uniformly $C^{1,\frac{1}{6}}$ continuous up to the sonic curve and the sonic
curve is also $C^{1,\frac{1}{6}}$ continuous.
