A global existence and uniqueness result for a class of hyperbolic operators

Abstract

In this paper a global existence and uniqueness theorem for the Cauchy problem related to the class of hyperbolic operators with double characteristics $P=D^2_{t} - D^2_{x_1} - (t+ \lambda - \alpha (x_1))^2 D^2_{x_2},$ depending on the parameter $\lambda$ in the half-space $\Omega = \mathbb {R}^2 \times ]0, + \infty [$ is proved. In the first part of the paper, a priori estimates in Sobolev spaces $W^{r,2}$, with $r$ negative, have been established. Then, the regularity is studied by using these spaces and by constructing a Green’s function in the half-plane for the operator of the waves.