In this paper we discuss the C∞
well-posedness for second order hyperbolic equations
Pu=∂t2​u−a(t,x)∂x2​u=f with two independent
variables (t,x). Assuming that the C∞ function
a(t,x)≥0 verifies ∂tp​a(0,0)î€ =0 with some p
and that the discriminant Δ(x) of a(t,x) vanishes of
finite order at x=0, we prove that the Cauchy problem for P is
C∞ well-posed in a neighbourhood of the origin