Some well-posed Cauchy problem for second order hyperbolic equations with two independent variables

In this paper we discuss the $C^{\infty}$ well-posedness for second order hyperbolic equations $Pu=\partial_t^2u-a(t,x)\partial_x^2u=f$ with two independent variables $(t,x)$. Assuming that the $C^{\infty}$ function $a(t,x)\geq 0$ verifies $\partial_t^pa(0,0)\neq 0$ with some $p$ and that the discriminant $\Delta(x)$ of $a(t,x)$ vanishes of finite order at $x=0$, we prove that the Cauchy problem for $P$ is $C^{\infty}$ well-posed in a neighbourhood of the origin.