Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics

The Goursat problem of a mixed type equation (P - xi (2))P(xi xi) - 2 xi etaP(xi eta) + (P - eta (2)) P(eta eta) + 1/P (xi P(xi) + eta P(eta))(2) - 2(xi P(xi) + etaP(eta)) = 0, P greater than or equal to 0, is considered. At the ends of its supports we have P = 0, which means it is degenerate hyperbolic. We prove the global existence of a smooth solution to the degenerate Goursat problem up to a boundary where P = 0. This problem comes from the expansion of a wedge of gas with constant velocity into vacuum, in two-dimensional pressure-gradient equations in gas dynamics, where P is the pressure and P = 0 means vacuum.